/export/starexec/sandbox/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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YES

Problem:
 active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
 active(sqr(0())) -> mark(0())
 active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
 active(dbl(0())) -> mark(0())
 active(dbl(s(X))) -> mark(s(s(dbl(X))))
 active(add(0(),X)) -> mark(X)
 active(add(s(X),Y)) -> mark(s(add(X,Y)))
 active(first(0(),X)) -> mark(nil())
 active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
 active(half(0())) -> mark(0())
 active(half(s(0()))) -> mark(0())
 active(half(s(s(X)))) -> mark(s(half(X)))
 active(half(dbl(X))) -> mark(X)
 active(terms(X)) -> terms(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(recip(X)) -> recip(active(X))
 active(sqr(X)) -> sqr(active(X))
 active(s(X)) -> s(active(X))
 active(add(X1,X2)) -> add(active(X1),X2)
 active(add(X1,X2)) -> add(X1,active(X2))
 active(dbl(X)) -> dbl(active(X))
 active(first(X1,X2)) -> first(active(X1),X2)
 active(first(X1,X2)) -> first(X1,active(X2))
 active(half(X)) -> half(active(X))
 terms(mark(X)) -> mark(terms(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 recip(mark(X)) -> mark(recip(X))
 sqr(mark(X)) -> mark(sqr(X))
 s(mark(X)) -> mark(s(X))
 add(mark(X1),X2) -> mark(add(X1,X2))
 add(X1,mark(X2)) -> mark(add(X1,X2))
 dbl(mark(X)) -> mark(dbl(X))
 first(mark(X1),X2) -> mark(first(X1,X2))
 first(X1,mark(X2)) -> mark(first(X1,X2))
 half(mark(X)) -> mark(half(X))
 proper(terms(X)) -> terms(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(recip(X)) -> recip(proper(X))
 proper(sqr(X)) -> sqr(proper(X))
 proper(s(X)) -> s(proper(X))
 proper(0()) -> ok(0())
 proper(add(X1,X2)) -> add(proper(X1),proper(X2))
 proper(dbl(X)) -> dbl(proper(X))
 proper(first(X1,X2)) -> first(proper(X1),proper(X2))
 proper(nil()) -> ok(nil())
 proper(half(X)) -> half(proper(X))
 terms(ok(X)) -> ok(terms(X))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 recip(ok(X)) -> ok(recip(X))
 sqr(ok(X)) -> ok(sqr(X))
 s(ok(X)) -> ok(s(X))
 add(ok(X1),ok(X2)) -> ok(add(X1,X2))
 dbl(ok(X)) -> ok(dbl(X))
 first(ok(X1),ok(X2)) -> ok(first(X1,X2))
 half(ok(X)) -> ok(half(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Proof:
 DP Processor:
  DPs:
   active#(terms(N)) -> s#(N)
   active#(terms(N)) -> terms#(s(N))
   active#(terms(N)) -> sqr#(N)
   active#(terms(N)) -> recip#(sqr(N))
   active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N)))
   active#(sqr(s(X))) -> dbl#(X)
   active#(sqr(s(X))) -> sqr#(X)
   active#(sqr(s(X))) -> add#(sqr(X),dbl(X))
   active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X)))
   active#(dbl(s(X))) -> dbl#(X)
   active#(dbl(s(X))) -> s#(dbl(X))
   active#(dbl(s(X))) -> s#(s(dbl(X)))
   active#(add(s(X),Y)) -> add#(X,Y)
   active#(add(s(X),Y)) -> s#(add(X,Y))
   active#(first(s(X),cons(Y,Z))) -> first#(X,Z)
   active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z))
   active#(half(s(s(X)))) -> half#(X)
   active#(half(s(s(X)))) -> s#(half(X))
   active#(terms(X)) -> active#(X)
   active#(terms(X)) -> terms#(active(X))
   active#(cons(X1,X2)) -> active#(X1)
   active#(cons(X1,X2)) -> cons#(active(X1),X2)
   active#(recip(X)) -> active#(X)
   active#(recip(X)) -> recip#(active(X))
   active#(sqr(X)) -> active#(X)
   active#(sqr(X)) -> sqr#(active(X))
   active#(s(X)) -> active#(X)
   active#(s(X)) -> s#(active(X))
   active#(add(X1,X2)) -> active#(X1)
   active#(add(X1,X2)) -> add#(active(X1),X2)
   active#(add(X1,X2)) -> active#(X2)
   active#(add(X1,X2)) -> add#(X1,active(X2))
   active#(dbl(X)) -> active#(X)
   active#(dbl(X)) -> dbl#(active(X))
   active#(first(X1,X2)) -> active#(X1)
   active#(first(X1,X2)) -> first#(active(X1),X2)
   active#(first(X1,X2)) -> active#(X2)
   active#(first(X1,X2)) -> first#(X1,active(X2))
   active#(half(X)) -> active#(X)
   active#(half(X)) -> half#(active(X))
   terms#(mark(X)) -> terms#(X)
   cons#(mark(X1),X2) -> cons#(X1,X2)
   recip#(mark(X)) -> recip#(X)
   sqr#(mark(X)) -> sqr#(X)
   s#(mark(X)) -> s#(X)
   add#(mark(X1),X2) -> add#(X1,X2)
   add#(X1,mark(X2)) -> add#(X1,X2)
   dbl#(mark(X)) -> dbl#(X)
   first#(mark(X1),X2) -> first#(X1,X2)
   first#(X1,mark(X2)) -> first#(X1,X2)
   half#(mark(X)) -> half#(X)
   proper#(terms(X)) -> proper#(X)
   proper#(terms(X)) -> terms#(proper(X))
   proper#(cons(X1,X2)) -> proper#(X2)
   proper#(cons(X1,X2)) -> proper#(X1)
   proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
   proper#(recip(X)) -> proper#(X)
   proper#(recip(X)) -> recip#(proper(X))
   proper#(sqr(X)) -> proper#(X)
   proper#(sqr(X)) -> sqr#(proper(X))
   proper#(s(X)) -> proper#(X)
   proper#(s(X)) -> s#(proper(X))
   proper#(add(X1,X2)) -> proper#(X2)
   proper#(add(X1,X2)) -> proper#(X1)
   proper#(add(X1,X2)) -> add#(proper(X1),proper(X2))
   proper#(dbl(X)) -> proper#(X)
   proper#(dbl(X)) -> dbl#(proper(X))
   proper#(first(X1,X2)) -> proper#(X2)
   proper#(first(X1,X2)) -> proper#(X1)
   proper#(first(X1,X2)) -> first#(proper(X1),proper(X2))
   proper#(half(X)) -> proper#(X)
   proper#(half(X)) -> half#(proper(X))
   terms#(ok(X)) -> terms#(X)
   cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
   recip#(ok(X)) -> recip#(X)
   sqr#(ok(X)) -> sqr#(X)
   s#(ok(X)) -> s#(X)
   add#(ok(X1),ok(X2)) -> add#(X1,X2)
   dbl#(ok(X)) -> dbl#(X)
   first#(ok(X1),ok(X2)) -> first#(X1,X2)
   half#(ok(X)) -> half#(X)
   top#(mark(X)) -> proper#(X)
   top#(mark(X)) -> top#(proper(X))
   top#(ok(X)) -> active#(X)
   top#(ok(X)) -> top#(active(X))
  TRS:
   active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
   active(sqr(0())) -> mark(0())
   active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
   active(dbl(0())) -> mark(0())
   active(dbl(s(X))) -> mark(s(s(dbl(X))))
   active(add(0(),X)) -> mark(X)
   active(add(s(X),Y)) -> mark(s(add(X,Y)))
   active(first(0(),X)) -> mark(nil())
   active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
   active(half(0())) -> mark(0())
   active(half(s(0()))) -> mark(0())
   active(half(s(s(X)))) -> mark(s(half(X)))
   active(half(dbl(X))) -> mark(X)
   active(terms(X)) -> terms(active(X))
   active(cons(X1,X2)) -> cons(active(X1),X2)
   active(recip(X)) -> recip(active(X))
   active(sqr(X)) -> sqr(active(X))
   active(s(X)) -> s(active(X))
   active(add(X1,X2)) -> add(active(X1),X2)
   active(add(X1,X2)) -> add(X1,active(X2))
   active(dbl(X)) -> dbl(active(X))
   active(first(X1,X2)) -> first(active(X1),X2)
   active(first(X1,X2)) -> first(X1,active(X2))
   active(half(X)) -> half(active(X))
   terms(mark(X)) -> mark(terms(X))
   cons(mark(X1),X2) -> mark(cons(X1,X2))
   recip(mark(X)) -> mark(recip(X))
   sqr(mark(X)) -> mark(sqr(X))
   s(mark(X)) -> mark(s(X))
   add(mark(X1),X2) -> mark(add(X1,X2))
   add(X1,mark(X2)) -> mark(add(X1,X2))
   dbl(mark(X)) -> mark(dbl(X))
   first(mark(X1),X2) -> mark(first(X1,X2))
   first(X1,mark(X2)) -> mark(first(X1,X2))
   half(mark(X)) -> mark(half(X))
   proper(terms(X)) -> terms(proper(X))
   proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
   proper(recip(X)) -> recip(proper(X))
   proper(sqr(X)) -> sqr(proper(X))
   proper(s(X)) -> s(proper(X))
   proper(0()) -> ok(0())
   proper(add(X1,X2)) -> add(proper(X1),proper(X2))
   proper(dbl(X)) -> dbl(proper(X))
   proper(first(X1,X2)) -> first(proper(X1),proper(X2))
   proper(nil()) -> ok(nil())
   proper(half(X)) -> half(proper(X))
   terms(ok(X)) -> ok(terms(X))
   cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
   recip(ok(X)) -> ok(recip(X))
   sqr(ok(X)) -> ok(sqr(X))
   s(ok(X)) -> ok(s(X))
   add(ok(X1),ok(X2)) -> ok(add(X1,X2))
   dbl(ok(X)) -> ok(dbl(X))
   first(ok(X1),ok(X2)) -> ok(first(X1,X2))
   half(ok(X)) -> ok(half(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))
  TDG Processor:
   DPs:
    active#(terms(N)) -> s#(N)
    active#(terms(N)) -> terms#(s(N))
    active#(terms(N)) -> sqr#(N)
    active#(terms(N)) -> recip#(sqr(N))
    active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N)))
    active#(sqr(s(X))) -> dbl#(X)
    active#(sqr(s(X))) -> sqr#(X)
    active#(sqr(s(X))) -> add#(sqr(X),dbl(X))
    active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X)))
    active#(dbl(s(X))) -> dbl#(X)
    active#(dbl(s(X))) -> s#(dbl(X))
    active#(dbl(s(X))) -> s#(s(dbl(X)))
    active#(add(s(X),Y)) -> add#(X,Y)
    active#(add(s(X),Y)) -> s#(add(X,Y))
    active#(first(s(X),cons(Y,Z))) -> first#(X,Z)
    active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z))
    active#(half(s(s(X)))) -> half#(X)
    active#(half(s(s(X)))) -> s#(half(X))
    active#(terms(X)) -> active#(X)
    active#(terms(X)) -> terms#(active(X))
    active#(cons(X1,X2)) -> active#(X1)
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(recip(X)) -> active#(X)
    active#(recip(X)) -> recip#(active(X))
    active#(sqr(X)) -> active#(X)
    active#(sqr(X)) -> sqr#(active(X))
    active#(s(X)) -> active#(X)
    active#(s(X)) -> s#(active(X))
    active#(add(X1,X2)) -> active#(X1)
    active#(add(X1,X2)) -> add#(active(X1),X2)
    active#(add(X1,X2)) -> active#(X2)
    active#(add(X1,X2)) -> add#(X1,active(X2))
    active#(dbl(X)) -> active#(X)
    active#(dbl(X)) -> dbl#(active(X))
    active#(first(X1,X2)) -> active#(X1)
    active#(first(X1,X2)) -> first#(active(X1),X2)
    active#(first(X1,X2)) -> active#(X2)
    active#(first(X1,X2)) -> first#(X1,active(X2))
    active#(half(X)) -> active#(X)
    active#(half(X)) -> half#(active(X))
    terms#(mark(X)) -> terms#(X)
    cons#(mark(X1),X2) -> cons#(X1,X2)
    recip#(mark(X)) -> recip#(X)
    sqr#(mark(X)) -> sqr#(X)
    s#(mark(X)) -> s#(X)
    add#(mark(X1),X2) -> add#(X1,X2)
    add#(X1,mark(X2)) -> add#(X1,X2)
    dbl#(mark(X)) -> dbl#(X)
    first#(mark(X1),X2) -> first#(X1,X2)
    first#(X1,mark(X2)) -> first#(X1,X2)
    half#(mark(X)) -> half#(X)
    proper#(terms(X)) -> proper#(X)
    proper#(terms(X)) -> terms#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(recip(X)) -> proper#(X)
    proper#(recip(X)) -> recip#(proper(X))
    proper#(sqr(X)) -> proper#(X)
    proper#(sqr(X)) -> sqr#(proper(X))
    proper#(s(X)) -> proper#(X)
    proper#(s(X)) -> s#(proper(X))
    proper#(add(X1,X2)) -> proper#(X2)
    proper#(add(X1,X2)) -> proper#(X1)
    proper#(add(X1,X2)) -> add#(proper(X1),proper(X2))
    proper#(dbl(X)) -> proper#(X)
    proper#(dbl(X)) -> dbl#(proper(X))
    proper#(first(X1,X2)) -> proper#(X2)
    proper#(first(X1,X2)) -> proper#(X1)
    proper#(first(X1,X2)) -> first#(proper(X1),proper(X2))
    proper#(half(X)) -> proper#(X)
    proper#(half(X)) -> half#(proper(X))
    terms#(ok(X)) -> terms#(X)
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    recip#(ok(X)) -> recip#(X)
    sqr#(ok(X)) -> sqr#(X)
    s#(ok(X)) -> s#(X)
    add#(ok(X1),ok(X2)) -> add#(X1,X2)
    dbl#(ok(X)) -> dbl#(X)
    first#(ok(X1),ok(X2)) -> first#(X1,X2)
    half#(ok(X)) -> half#(X)
    top#(mark(X)) -> proper#(X)
    top#(mark(X)) -> top#(proper(X))
    top#(ok(X)) -> active#(X)
    top#(ok(X)) -> top#(active(X))
   TRS:
    active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
    active(sqr(0())) -> mark(0())
    active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
    active(dbl(0())) -> mark(0())
    active(dbl(s(X))) -> mark(s(s(dbl(X))))
    active(add(0(),X)) -> mark(X)
    active(add(s(X),Y)) -> mark(s(add(X,Y)))
    active(first(0(),X)) -> mark(nil())
    active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
    active(half(0())) -> mark(0())
    active(half(s(0()))) -> mark(0())
    active(half(s(s(X)))) -> mark(s(half(X)))
    active(half(dbl(X))) -> mark(X)
    active(terms(X)) -> terms(active(X))
    active(cons(X1,X2)) -> cons(active(X1),X2)
    active(recip(X)) -> recip(active(X))
    active(sqr(X)) -> sqr(active(X))
    active(s(X)) -> s(active(X))
    active(add(X1,X2)) -> add(active(X1),X2)
    active(add(X1,X2)) -> add(X1,active(X2))
    active(dbl(X)) -> dbl(active(X))
    active(first(X1,X2)) -> first(active(X1),X2)
    active(first(X1,X2)) -> first(X1,active(X2))
    active(half(X)) -> half(active(X))
    terms(mark(X)) -> mark(terms(X))
    cons(mark(X1),X2) -> mark(cons(X1,X2))
    recip(mark(X)) -> mark(recip(X))
    sqr(mark(X)) -> mark(sqr(X))
    s(mark(X)) -> mark(s(X))
    add(mark(X1),X2) -> mark(add(X1,X2))
    add(X1,mark(X2)) -> mark(add(X1,X2))
    dbl(mark(X)) -> mark(dbl(X))
    first(mark(X1),X2) -> mark(first(X1,X2))
    first(X1,mark(X2)) -> mark(first(X1,X2))
    half(mark(X)) -> mark(half(X))
    proper(terms(X)) -> terms(proper(X))
    proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
    proper(recip(X)) -> recip(proper(X))
    proper(sqr(X)) -> sqr(proper(X))
    proper(s(X)) -> s(proper(X))
    proper(0()) -> ok(0())
    proper(add(X1,X2)) -> add(proper(X1),proper(X2))
    proper(dbl(X)) -> dbl(proper(X))
    proper(first(X1,X2)) -> first(proper(X1),proper(X2))
    proper(nil()) -> ok(nil())
    proper(half(X)) -> half(proper(X))
    terms(ok(X)) -> ok(terms(X))
    cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
    recip(ok(X)) -> ok(recip(X))
    sqr(ok(X)) -> ok(sqr(X))
    s(ok(X)) -> ok(s(X))
    add(ok(X1),ok(X2)) -> ok(add(X1,X2))
    dbl(ok(X)) -> ok(dbl(X))
    first(ok(X1),ok(X2)) -> ok(first(X1,X2))
    half(ok(X)) -> ok(half(X))
    top(mark(X)) -> top(proper(X))
    top(ok(X)) -> top(active(X))
   graph:
    top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> top#(active(X))
    top#(ok(X)) -> top#(active(X)) -> top#(ok(X)) -> active#(X)
    top#(ok(X)) -> top#(active(X)) ->
    top#(mark(X)) -> top#(proper(X))
    top#(ok(X)) -> top#(active(X)) -> top#(mark(X)) -> proper#(X)
    top#(ok(X)) -> active#(X) -> active#(half(X)) -> half#(active(X))
    top#(ok(X)) -> active#(X) -> active#(half(X)) -> active#(X)
    top#(ok(X)) -> active#(X) ->
    active#(first(X1,X2)) -> first#(X1,active(X2))
    top#(ok(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X2)
    top#(ok(X)) -> active#(X) ->
    active#(first(X1,X2)) -> first#(active(X1),X2)
    top#(ok(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X1)
    top#(ok(X)) -> active#(X) -> active#(dbl(X)) -> dbl#(active(X))
    top#(ok(X)) -> active#(X) -> active#(dbl(X)) -> active#(X)
    top#(ok(X)) -> active#(X) ->
    active#(add(X1,X2)) -> add#(X1,active(X2))
    top#(ok(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X2)
    top#(ok(X)) -> active#(X) ->
    active#(add(X1,X2)) -> add#(active(X1),X2)
    top#(ok(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X1)
    top#(ok(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
    top#(ok(X)) -> active#(X) -> active#(s(X)) -> active#(X)
    top#(ok(X)) -> active#(X) -> active#(sqr(X)) -> sqr#(active(X))
    top#(ok(X)) -> active#(X) -> active#(sqr(X)) -> active#(X)
    top#(ok(X)) -> active#(X) -> active#(recip(X)) -> recip#(active(X))
    top#(ok(X)) -> active#(X) -> active#(recip(X)) -> active#(X)
    top#(ok(X)) -> active#(X) ->
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    top#(ok(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1)
    top#(ok(X)) -> active#(X) -> active#(terms(X)) -> terms#(active(X))
    top#(ok(X)) -> active#(X) -> active#(terms(X)) -> active#(X)
    top#(ok(X)) -> active#(X) -> active#(half(s(s(X)))) -> s#(half(X))
    top#(ok(X)) -> active#(X) -> active#(half(s(s(X)))) -> half#(X)
    top#(ok(X)) -> active#(X) ->
    active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z))
    top#(ok(X)) -> active#(X) ->
    active#(first(s(X),cons(Y,Z))) -> first#(X,Z)
    top#(ok(X)) -> active#(X) -> active#(add(s(X),Y)) -> s#(add(X,Y))
    top#(ok(X)) -> active#(X) -> active#(add(s(X),Y)) -> add#(X,Y)
    top#(ok(X)) -> active#(X) -> active#(dbl(s(X))) -> s#(s(dbl(X)))
    top#(ok(X)) -> active#(X) -> active#(dbl(s(X))) -> s#(dbl(X))
    top#(ok(X)) -> active#(X) -> active#(dbl(s(X))) -> dbl#(X)
    top#(ok(X)) -> active#(X) ->
    active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X)))
    top#(ok(X)) -> active#(X) -> active#(sqr(s(X))) -> add#(sqr(X),dbl(X))
    top#(ok(X)) -> active#(X) -> active#(sqr(s(X))) -> sqr#(X)
    top#(ok(X)) -> active#(X) -> active#(sqr(s(X))) -> dbl#(X)
    top#(ok(X)) -> active#(X) ->
    active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N)))
    top#(ok(X)) -> active#(X) -> active#(terms(N)) -> recip#(sqr(N))
    top#(ok(X)) -> active#(X) -> active#(terms(N)) -> sqr#(N)
    top#(ok(X)) -> active#(X) -> active#(terms(N)) -> terms#(s(N))
    top#(ok(X)) -> active#(X) -> active#(terms(N)) -> s#(N)
    top#(mark(X)) -> top#(proper(X)) ->
    top#(ok(X)) -> top#(active(X))
    top#(mark(X)) -> top#(proper(X)) -> top#(ok(X)) -> active#(X)
    top#(mark(X)) -> top#(proper(X)) ->
    top#(mark(X)) -> top#(proper(X))
    top#(mark(X)) -> top#(proper(X)) -> top#(mark(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) -> proper#(half(X)) -> half#(proper(X))
    top#(mark(X)) -> proper#(X) -> proper#(half(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) ->
    proper#(first(X1,X2)) -> first#(proper(X1),proper(X2))
    top#(mark(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X1)
    top#(mark(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X2)
    top#(mark(X)) -> proper#(X) -> proper#(dbl(X)) -> dbl#(proper(X))
    top#(mark(X)) -> proper#(X) -> proper#(dbl(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) ->
    proper#(add(X1,X2)) -> add#(proper(X1),proper(X2))
    top#(mark(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X1)
    top#(mark(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X2)
    top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
    top#(mark(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) -> proper#(sqr(X)) -> sqr#(proper(X))
    top#(mark(X)) -> proper#(X) -> proper#(sqr(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) ->
    proper#(recip(X)) -> recip#(proper(X))
    top#(mark(X)) -> proper#(X) -> proper#(recip(X)) -> proper#(X)
    top#(mark(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1)
    top#(mark(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2)
    top#(mark(X)) -> proper#(X) ->
    proper#(terms(X)) -> terms#(proper(X))
    top#(mark(X)) -> proper#(X) -> proper#(terms(X)) -> proper#(X)
    proper#(half(X)) -> proper#(X) ->
    proper#(half(X)) -> half#(proper(X))
    proper#(half(X)) -> proper#(X) -> proper#(half(X)) -> proper#(X)
    proper#(half(X)) -> proper#(X) ->
    proper#(first(X1,X2)) -> first#(proper(X1),proper(X2))
    proper#(half(X)) -> proper#(X) ->
    proper#(first(X1,X2)) -> proper#(X1)
    proper#(half(X)) -> proper#(X) ->
    proper#(first(X1,X2)) -> proper#(X2)
    proper#(half(X)) -> proper#(X) ->
    proper#(dbl(X)) -> dbl#(proper(X))
    proper#(half(X)) -> proper#(X) -> proper#(dbl(X)) -> proper#(X)
    proper#(half(X)) -> proper#(X) ->
    proper#(add(X1,X2)) -> add#(proper(X1),proper(X2))
    proper#(half(X)) -> proper#(X) ->
    proper#(add(X1,X2)) -> proper#(X1)
    proper#(half(X)) -> proper#(X) ->
    proper#(add(X1,X2)) -> proper#(X2)
    proper#(half(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
    proper#(half(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    proper#(half(X)) -> proper#(X) ->
    proper#(sqr(X)) -> sqr#(proper(X))
    proper#(half(X)) -> proper#(X) -> proper#(sqr(X)) -> proper#(X)
    proper#(half(X)) -> proper#(X) ->
    proper#(recip(X)) -> recip#(proper(X))
    proper#(half(X)) -> proper#(X) -> proper#(recip(X)) -> proper#(X)
    proper#(half(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(half(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(half(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(half(X)) -> proper#(X) ->
    proper#(terms(X)) -> terms#(proper(X))
    proper#(half(X)) -> proper#(X) ->
    proper#(terms(X)) -> proper#(X)
    proper#(half(X)) -> half#(proper(X)) ->
    half#(ok(X)) -> half#(X)
    proper#(half(X)) -> half#(proper(X)) ->
    half#(mark(X)) -> half#(X)
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(half(X)) -> half#(proper(X))
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(half(X)) -> proper#(X)
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(first(X1,X2)) -> first#(proper(X1),proper(X2))
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(first(X1,X2)) -> proper#(X1)
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(first(X1,X2)) -> proper#(X2)
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(dbl(X)) -> dbl#(proper(X))
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(dbl(X)) -> proper#(X)
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(add(X1,X2)) -> add#(proper(X1),proper(X2))
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(add(X1,X2)) -> proper#(X1)
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(add(X1,X2)) -> proper#(X2)
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> proper#(X)
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(sqr(X)) -> sqr#(proper(X))
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(sqr(X)) -> proper#(X)
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(recip(X)) -> recip#(proper(X))
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(recip(X)) -> proper#(X)
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(terms(X)) -> terms#(proper(X))
    proper#(first(X1,X2)) -> proper#(X2) ->
    proper#(terms(X)) -> proper#(X)
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(half(X)) -> half#(proper(X))
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(half(X)) -> proper#(X)
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(first(X1,X2)) -> first#(proper(X1),proper(X2))
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(first(X1,X2)) -> proper#(X1)
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(first(X1,X2)) -> proper#(X2)
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(dbl(X)) -> dbl#(proper(X))
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(dbl(X)) -> proper#(X)
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(add(X1,X2)) -> add#(proper(X1),proper(X2))
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(add(X1,X2)) -> proper#(X1)
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(add(X1,X2)) -> proper#(X2)
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> proper#(X)
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(sqr(X)) -> sqr#(proper(X))
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(sqr(X)) -> proper#(X)
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(recip(X)) -> recip#(proper(X))
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(recip(X)) -> proper#(X)
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(terms(X)) -> terms#(proper(X))
    proper#(first(X1,X2)) -> proper#(X1) ->
    proper#(terms(X)) -> proper#(X)
    proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) ->
    first#(ok(X1),ok(X2)) -> first#(X1,X2)
    proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) ->
    first#(X1,mark(X2)) -> first#(X1,X2)
    proper#(first(X1,X2)) -> first#(proper(X1),proper(X2)) ->
    first#(mark(X1),X2) -> first#(X1,X2)
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(half(X)) -> half#(proper(X))
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(half(X)) -> proper#(X)
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(first(X1,X2)) -> first#(proper(X1),proper(X2))
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(first(X1,X2)) -> proper#(X1)
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(first(X1,X2)) -> proper#(X2)
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(dbl(X)) -> dbl#(proper(X))
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(dbl(X)) -> proper#(X)
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(add(X1,X2)) -> add#(proper(X1),proper(X2))
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(add(X1,X2)) -> proper#(X1)
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(add(X1,X2)) -> proper#(X2)
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> proper#(X)
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(sqr(X)) -> sqr#(proper(X))
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(sqr(X)) -> proper#(X)
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(recip(X)) -> recip#(proper(X))
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(recip(X)) -> proper#(X)
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(terms(X)) -> terms#(proper(X))
    proper#(add(X1,X2)) -> proper#(X2) ->
    proper#(terms(X)) -> proper#(X)
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(half(X)) -> half#(proper(X))
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(half(X)) -> proper#(X)
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(first(X1,X2)) -> first#(proper(X1),proper(X2))
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(first(X1,X2)) -> proper#(X1)
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(first(X1,X2)) -> proper#(X2)
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(dbl(X)) -> dbl#(proper(X))
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(dbl(X)) -> proper#(X)
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(add(X1,X2)) -> add#(proper(X1),proper(X2))
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(add(X1,X2)) -> proper#(X1)
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(add(X1,X2)) -> proper#(X2)
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> proper#(X)
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(sqr(X)) -> sqr#(proper(X))
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(sqr(X)) -> proper#(X)
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(recip(X)) -> recip#(proper(X))
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(recip(X)) -> proper#(X)
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(terms(X)) -> terms#(proper(X))
    proper#(add(X1,X2)) -> proper#(X1) ->
    proper#(terms(X)) -> proper#(X)
    proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) ->
    add#(ok(X1),ok(X2)) -> add#(X1,X2)
    proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) ->
    add#(X1,mark(X2)) -> add#(X1,X2)
    proper#(add(X1,X2)) -> add#(proper(X1),proper(X2)) ->
    add#(mark(X1),X2) -> add#(X1,X2)
    proper#(dbl(X)) -> proper#(X) ->
    proper#(half(X)) -> half#(proper(X))
    proper#(dbl(X)) -> proper#(X) -> proper#(half(X)) -> proper#(X)
    proper#(dbl(X)) -> proper#(X) ->
    proper#(first(X1,X2)) -> first#(proper(X1),proper(X2))
    proper#(dbl(X)) -> proper#(X) ->
    proper#(first(X1,X2)) -> proper#(X1)
    proper#(dbl(X)) -> proper#(X) ->
    proper#(first(X1,X2)) -> proper#(X2)
    proper#(dbl(X)) -> proper#(X) ->
    proper#(dbl(X)) -> dbl#(proper(X))
    proper#(dbl(X)) -> proper#(X) -> proper#(dbl(X)) -> proper#(X)
    proper#(dbl(X)) -> proper#(X) ->
    proper#(add(X1,X2)) -> add#(proper(X1),proper(X2))
    proper#(dbl(X)) -> proper#(X) ->
    proper#(add(X1,X2)) -> proper#(X1)
    proper#(dbl(X)) -> proper#(X) ->
    proper#(add(X1,X2)) -> proper#(X2)
    proper#(dbl(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
    proper#(dbl(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    proper#(dbl(X)) -> proper#(X) ->
    proper#(sqr(X)) -> sqr#(proper(X))
    proper#(dbl(X)) -> proper#(X) -> proper#(sqr(X)) -> proper#(X)
    proper#(dbl(X)) -> proper#(X) ->
    proper#(recip(X)) -> recip#(proper(X))
    proper#(dbl(X)) -> proper#(X) -> proper#(recip(X)) -> proper#(X)
    proper#(dbl(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(dbl(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(dbl(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(dbl(X)) -> proper#(X) ->
    proper#(terms(X)) -> terms#(proper(X))
    proper#(dbl(X)) -> proper#(X) ->
    proper#(terms(X)) -> proper#(X)
    proper#(dbl(X)) -> dbl#(proper(X)) -> dbl#(ok(X)) -> dbl#(X)
    proper#(dbl(X)) -> dbl#(proper(X)) ->
    dbl#(mark(X)) -> dbl#(X)
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(half(X)) -> half#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(half(X)) -> proper#(X)
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(first(X1,X2)) -> first#(proper(X1),proper(X2))
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(first(X1,X2)) -> proper#(X1)
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(first(X1,X2)) -> proper#(X2)
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(dbl(X)) -> dbl#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(dbl(X)) -> proper#(X)
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(add(X1,X2)) -> add#(proper(X1),proper(X2))
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(add(X1,X2)) -> proper#(X1)
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(add(X1,X2)) -> proper#(X2)
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(s(X)) -> proper#(X)
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(sqr(X)) -> sqr#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(sqr(X)) -> proper#(X)
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(recip(X)) -> recip#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(recip(X)) -> proper#(X)
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(terms(X)) -> terms#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X2) ->
    proper#(terms(X)) -> proper#(X)
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(half(X)) -> half#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(half(X)) -> proper#(X)
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(first(X1,X2)) -> first#(proper(X1),proper(X2))
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(first(X1,X2)) -> proper#(X1)
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(first(X1,X2)) -> proper#(X2)
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(dbl(X)) -> dbl#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(dbl(X)) -> proper#(X)
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(add(X1,X2)) -> add#(proper(X1),proper(X2))
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(add(X1,X2)) -> proper#(X1)
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(add(X1,X2)) -> proper#(X2)
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(s(X)) -> proper#(X)
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(sqr(X)) -> sqr#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(sqr(X)) -> proper#(X)
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(recip(X)) -> recip#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(recip(X)) -> proper#(X)
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(terms(X)) -> terms#(proper(X))
    proper#(cons(X1,X2)) -> proper#(X1) ->
    proper#(terms(X)) -> proper#(X)
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) ->
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2)) ->
    cons#(mark(X1),X2) -> cons#(X1,X2)
    proper#(s(X)) -> proper#(X) -> proper#(half(X)) -> half#(proper(X))
    proper#(s(X)) -> proper#(X) -> proper#(half(X)) -> proper#(X)
    proper#(s(X)) -> proper#(X) ->
    proper#(first(X1,X2)) -> first#(proper(X1),proper(X2))
    proper#(s(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X1)
    proper#(s(X)) -> proper#(X) -> proper#(first(X1,X2)) -> proper#(X2)
    proper#(s(X)) -> proper#(X) -> proper#(dbl(X)) -> dbl#(proper(X))
    proper#(s(X)) -> proper#(X) -> proper#(dbl(X)) -> proper#(X)
    proper#(s(X)) -> proper#(X) ->
    proper#(add(X1,X2)) -> add#(proper(X1),proper(X2))
    proper#(s(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X1)
    proper#(s(X)) -> proper#(X) -> proper#(add(X1,X2)) -> proper#(X2)
    proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
    proper#(s(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    proper#(s(X)) -> proper#(X) -> proper#(sqr(X)) -> sqr#(proper(X))
    proper#(s(X)) -> proper#(X) -> proper#(sqr(X)) -> proper#(X)
    proper#(s(X)) -> proper#(X) ->
    proper#(recip(X)) -> recip#(proper(X))
    proper#(s(X)) -> proper#(X) -> proper#(recip(X)) -> proper#(X)
    proper#(s(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X1)
    proper#(s(X)) -> proper#(X) -> proper#(cons(X1,X2)) -> proper#(X2)
    proper#(s(X)) -> proper#(X) ->
    proper#(terms(X)) -> terms#(proper(X))
    proper#(s(X)) -> proper#(X) -> proper#(terms(X)) -> proper#(X)
    proper#(s(X)) -> s#(proper(X)) -> s#(ok(X)) -> s#(X)
    proper#(s(X)) -> s#(proper(X)) -> s#(mark(X)) -> s#(X)
    proper#(recip(X)) -> proper#(X) ->
    proper#(half(X)) -> half#(proper(X))
    proper#(recip(X)) -> proper#(X) ->
    proper#(half(X)) -> proper#(X)
    proper#(recip(X)) -> proper#(X) ->
    proper#(first(X1,X2)) -> first#(proper(X1),proper(X2))
    proper#(recip(X)) -> proper#(X) ->
    proper#(first(X1,X2)) -> proper#(X1)
    proper#(recip(X)) -> proper#(X) ->
    proper#(first(X1,X2)) -> proper#(X2)
    proper#(recip(X)) -> proper#(X) ->
    proper#(dbl(X)) -> dbl#(proper(X))
    proper#(recip(X)) -> proper#(X) -> proper#(dbl(X)) -> proper#(X)
    proper#(recip(X)) -> proper#(X) ->
    proper#(add(X1,X2)) -> add#(proper(X1),proper(X2))
    proper#(recip(X)) -> proper#(X) ->
    proper#(add(X1,X2)) -> proper#(X1)
    proper#(recip(X)) -> proper#(X) ->
    proper#(add(X1,X2)) -> proper#(X2)
    proper#(recip(X)) -> proper#(X) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(recip(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    proper#(recip(X)) -> proper#(X) ->
    proper#(sqr(X)) -> sqr#(proper(X))
    proper#(recip(X)) -> proper#(X) -> proper#(sqr(X)) -> proper#(X)
    proper#(recip(X)) -> proper#(X) ->
    proper#(recip(X)) -> recip#(proper(X))
    proper#(recip(X)) -> proper#(X) ->
    proper#(recip(X)) -> proper#(X)
    proper#(recip(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(recip(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(recip(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(recip(X)) -> proper#(X) ->
    proper#(terms(X)) -> terms#(proper(X))
    proper#(recip(X)) -> proper#(X) ->
    proper#(terms(X)) -> proper#(X)
    proper#(recip(X)) -> recip#(proper(X)) ->
    recip#(ok(X)) -> recip#(X)
    proper#(recip(X)) -> recip#(proper(X)) ->
    recip#(mark(X)) -> recip#(X)
    proper#(sqr(X)) -> proper#(X) ->
    proper#(half(X)) -> half#(proper(X))
    proper#(sqr(X)) -> proper#(X) -> proper#(half(X)) -> proper#(X)
    proper#(sqr(X)) -> proper#(X) ->
    proper#(first(X1,X2)) -> first#(proper(X1),proper(X2))
    proper#(sqr(X)) -> proper#(X) ->
    proper#(first(X1,X2)) -> proper#(X1)
    proper#(sqr(X)) -> proper#(X) ->
    proper#(first(X1,X2)) -> proper#(X2)
    proper#(sqr(X)) -> proper#(X) ->
    proper#(dbl(X)) -> dbl#(proper(X))
    proper#(sqr(X)) -> proper#(X) -> proper#(dbl(X)) -> proper#(X)
    proper#(sqr(X)) -> proper#(X) ->
    proper#(add(X1,X2)) -> add#(proper(X1),proper(X2))
    proper#(sqr(X)) -> proper#(X) ->
    proper#(add(X1,X2)) -> proper#(X1)
    proper#(sqr(X)) -> proper#(X) ->
    proper#(add(X1,X2)) -> proper#(X2)
    proper#(sqr(X)) -> proper#(X) -> proper#(s(X)) -> s#(proper(X))
    proper#(sqr(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    proper#(sqr(X)) -> proper#(X) ->
    proper#(sqr(X)) -> sqr#(proper(X))
    proper#(sqr(X)) -> proper#(X) -> proper#(sqr(X)) -> proper#(X)
    proper#(sqr(X)) -> proper#(X) ->
    proper#(recip(X)) -> recip#(proper(X))
    proper#(sqr(X)) -> proper#(X) -> proper#(recip(X)) -> proper#(X)
    proper#(sqr(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(sqr(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(sqr(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(sqr(X)) -> proper#(X) ->
    proper#(terms(X)) -> terms#(proper(X))
    proper#(sqr(X)) -> proper#(X) ->
    proper#(terms(X)) -> proper#(X)
    proper#(sqr(X)) -> sqr#(proper(X)) -> sqr#(ok(X)) -> sqr#(X)
    proper#(sqr(X)) -> sqr#(proper(X)) -> sqr#(mark(X)) -> sqr#(X)
    proper#(terms(X)) -> proper#(X) ->
    proper#(half(X)) -> half#(proper(X))
    proper#(terms(X)) -> proper#(X) ->
    proper#(half(X)) -> proper#(X)
    proper#(terms(X)) -> proper#(X) ->
    proper#(first(X1,X2)) -> first#(proper(X1),proper(X2))
    proper#(terms(X)) -> proper#(X) ->
    proper#(first(X1,X2)) -> proper#(X1)
    proper#(terms(X)) -> proper#(X) ->
    proper#(first(X1,X2)) -> proper#(X2)
    proper#(terms(X)) -> proper#(X) ->
    proper#(dbl(X)) -> dbl#(proper(X))
    proper#(terms(X)) -> proper#(X) -> proper#(dbl(X)) -> proper#(X)
    proper#(terms(X)) -> proper#(X) ->
    proper#(add(X1,X2)) -> add#(proper(X1),proper(X2))
    proper#(terms(X)) -> proper#(X) ->
    proper#(add(X1,X2)) -> proper#(X1)
    proper#(terms(X)) -> proper#(X) ->
    proper#(add(X1,X2)) -> proper#(X2)
    proper#(terms(X)) -> proper#(X) ->
    proper#(s(X)) -> s#(proper(X))
    proper#(terms(X)) -> proper#(X) -> proper#(s(X)) -> proper#(X)
    proper#(terms(X)) -> proper#(X) ->
    proper#(sqr(X)) -> sqr#(proper(X))
    proper#(terms(X)) -> proper#(X) -> proper#(sqr(X)) -> proper#(X)
    proper#(terms(X)) -> proper#(X) ->
    proper#(recip(X)) -> recip#(proper(X))
    proper#(terms(X)) -> proper#(X) ->
    proper#(recip(X)) -> proper#(X)
    proper#(terms(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> cons#(proper(X1),proper(X2))
    proper#(terms(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> proper#(X1)
    proper#(terms(X)) -> proper#(X) ->
    proper#(cons(X1,X2)) -> proper#(X2)
    proper#(terms(X)) -> proper#(X) ->
    proper#(terms(X)) -> terms#(proper(X))
    proper#(terms(X)) -> proper#(X) ->
    proper#(terms(X)) -> proper#(X)
    proper#(terms(X)) -> terms#(proper(X)) ->
    terms#(ok(X)) -> terms#(X)
    proper#(terms(X)) -> terms#(proper(X)) -> terms#(mark(X)) -> terms#(X)
    half#(ok(X)) -> half#(X) -> half#(ok(X)) -> half#(X)
    half#(ok(X)) -> half#(X) -> half#(mark(X)) -> half#(X)
    half#(mark(X)) -> half#(X) -> half#(ok(X)) -> half#(X)
    half#(mark(X)) -> half#(X) -> half#(mark(X)) -> half#(X)
    first#(ok(X1),ok(X2)) -> first#(X1,X2) ->
    first#(ok(X1),ok(X2)) -> first#(X1,X2)
    first#(ok(X1),ok(X2)) -> first#(X1,X2) ->
    first#(X1,mark(X2)) -> first#(X1,X2)
    first#(ok(X1),ok(X2)) -> first#(X1,X2) ->
    first#(mark(X1),X2) -> first#(X1,X2)
    first#(mark(X1),X2) -> first#(X1,X2) ->
    first#(ok(X1),ok(X2)) -> first#(X1,X2)
    first#(mark(X1),X2) -> first#(X1,X2) ->
    first#(X1,mark(X2)) -> first#(X1,X2)
    first#(mark(X1),X2) -> first#(X1,X2) ->
    first#(mark(X1),X2) -> first#(X1,X2)
    first#(X1,mark(X2)) -> first#(X1,X2) ->
    first#(ok(X1),ok(X2)) -> first#(X1,X2)
    first#(X1,mark(X2)) -> first#(X1,X2) ->
    first#(X1,mark(X2)) -> first#(X1,X2)
    first#(X1,mark(X2)) -> first#(X1,X2) ->
    first#(mark(X1),X2) -> first#(X1,X2)
    add#(ok(X1),ok(X2)) -> add#(X1,X2) ->
    add#(ok(X1),ok(X2)) -> add#(X1,X2)
    add#(ok(X1),ok(X2)) -> add#(X1,X2) ->
    add#(X1,mark(X2)) -> add#(X1,X2)
    add#(ok(X1),ok(X2)) -> add#(X1,X2) ->
    add#(mark(X1),X2) -> add#(X1,X2)
    add#(mark(X1),X2) -> add#(X1,X2) ->
    add#(ok(X1),ok(X2)) -> add#(X1,X2)
    add#(mark(X1),X2) -> add#(X1,X2) ->
    add#(X1,mark(X2)) -> add#(X1,X2)
    add#(mark(X1),X2) -> add#(X1,X2) ->
    add#(mark(X1),X2) -> add#(X1,X2)
    add#(X1,mark(X2)) -> add#(X1,X2) ->
    add#(ok(X1),ok(X2)) -> add#(X1,X2)
    add#(X1,mark(X2)) -> add#(X1,X2) ->
    add#(X1,mark(X2)) -> add#(X1,X2)
    add#(X1,mark(X2)) -> add#(X1,X2) -> add#(mark(X1),X2) -> add#(X1,X2)
    dbl#(ok(X)) -> dbl#(X) -> dbl#(ok(X)) -> dbl#(X)
    dbl#(ok(X)) -> dbl#(X) -> dbl#(mark(X)) -> dbl#(X)
    dbl#(mark(X)) -> dbl#(X) -> dbl#(ok(X)) -> dbl#(X)
    dbl#(mark(X)) -> dbl#(X) -> dbl#(mark(X)) -> dbl#(X)
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2) ->
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2) ->
    cons#(mark(X1),X2) -> cons#(X1,X2)
    cons#(mark(X1),X2) -> cons#(X1,X2) ->
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    cons#(mark(X1),X2) -> cons#(X1,X2) ->
    cons#(mark(X1),X2) -> cons#(X1,X2)
    recip#(ok(X)) -> recip#(X) -> recip#(ok(X)) -> recip#(X)
    recip#(ok(X)) -> recip#(X) -> recip#(mark(X)) -> recip#(X)
    recip#(mark(X)) -> recip#(X) -> recip#(ok(X)) -> recip#(X)
    recip#(mark(X)) -> recip#(X) -> recip#(mark(X)) -> recip#(X)
    sqr#(ok(X)) -> sqr#(X) -> sqr#(ok(X)) -> sqr#(X)
    sqr#(ok(X)) -> sqr#(X) -> sqr#(mark(X)) -> sqr#(X)
    sqr#(mark(X)) -> sqr#(X) -> sqr#(ok(X)) -> sqr#(X)
    sqr#(mark(X)) -> sqr#(X) -> sqr#(mark(X)) -> sqr#(X)
    terms#(ok(X)) -> terms#(X) -> terms#(ok(X)) -> terms#(X)
    terms#(ok(X)) -> terms#(X) -> terms#(mark(X)) -> terms#(X)
    terms#(mark(X)) -> terms#(X) -> terms#(ok(X)) -> terms#(X)
    terms#(mark(X)) -> terms#(X) -> terms#(mark(X)) -> terms#(X)
    s#(ok(X)) -> s#(X) -> s#(ok(X)) -> s#(X)
    s#(ok(X)) -> s#(X) -> s#(mark(X)) -> s#(X)
    s#(mark(X)) -> s#(X) -> s#(ok(X)) -> s#(X)
    s#(mark(X)) -> s#(X) -> s#(mark(X)) -> s#(X)
    active#(half(s(s(X)))) -> half#(X) ->
    half#(ok(X)) -> half#(X)
    active#(half(s(s(X)))) -> half#(X) ->
    half#(mark(X)) -> half#(X)
    active#(half(s(s(X)))) -> s#(half(X)) ->
    s#(ok(X)) -> s#(X)
    active#(half(s(s(X)))) -> s#(half(X)) ->
    s#(mark(X)) -> s#(X)
    active#(half(X)) -> half#(active(X)) ->
    half#(ok(X)) -> half#(X)
    active#(half(X)) -> half#(active(X)) ->
    half#(mark(X)) -> half#(X)
    active#(half(X)) -> active#(X) ->
    active#(half(X)) -> half#(active(X))
    active#(half(X)) -> active#(X) -> active#(half(X)) -> active#(X)
    active#(half(X)) -> active#(X) ->
    active#(first(X1,X2)) -> first#(X1,active(X2))
    active#(half(X)) -> active#(X) ->
    active#(first(X1,X2)) -> active#(X2)
    active#(half(X)) -> active#(X) ->
    active#(first(X1,X2)) -> first#(active(X1),X2)
    active#(half(X)) -> active#(X) ->
    active#(first(X1,X2)) -> active#(X1)
    active#(half(X)) -> active#(X) ->
    active#(dbl(X)) -> dbl#(active(X))
    active#(half(X)) -> active#(X) -> active#(dbl(X)) -> active#(X)
    active#(half(X)) -> active#(X) ->
    active#(add(X1,X2)) -> add#(X1,active(X2))
    active#(half(X)) -> active#(X) ->
    active#(add(X1,X2)) -> active#(X2)
    active#(half(X)) -> active#(X) ->
    active#(add(X1,X2)) -> add#(active(X1),X2)
    active#(half(X)) -> active#(X) ->
    active#(add(X1,X2)) -> active#(X1)
    active#(half(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
    active#(half(X)) -> active#(X) -> active#(s(X)) -> active#(X)
    active#(half(X)) -> active#(X) ->
    active#(sqr(X)) -> sqr#(active(X))
    active#(half(X)) -> active#(X) -> active#(sqr(X)) -> active#(X)
    active#(half(X)) -> active#(X) ->
    active#(recip(X)) -> recip#(active(X))
    active#(half(X)) -> active#(X) -> active#(recip(X)) -> active#(X)
    active#(half(X)) -> active#(X) ->
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(half(X)) -> active#(X) ->
    active#(cons(X1,X2)) -> active#(X1)
    active#(half(X)) -> active#(X) ->
    active#(terms(X)) -> terms#(active(X))
    active#(half(X)) -> active#(X) -> active#(terms(X)) -> active#(X)
    active#(half(X)) -> active#(X) ->
    active#(half(s(s(X)))) -> s#(half(X))
    active#(half(X)) -> active#(X) ->
    active#(half(s(s(X)))) -> half#(X)
    active#(half(X)) -> active#(X) ->
    active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z))
    active#(half(X)) -> active#(X) ->
    active#(first(s(X),cons(Y,Z))) -> first#(X,Z)
    active#(half(X)) -> active#(X) ->
    active#(add(s(X),Y)) -> s#(add(X,Y))
    active#(half(X)) -> active#(X) ->
    active#(add(s(X),Y)) -> add#(X,Y)
    active#(half(X)) -> active#(X) ->
    active#(dbl(s(X))) -> s#(s(dbl(X)))
    active#(half(X)) -> active#(X) ->
    active#(dbl(s(X))) -> s#(dbl(X))
    active#(half(X)) -> active#(X) -> active#(dbl(s(X))) -> dbl#(X)
    active#(half(X)) -> active#(X) ->
    active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X)))
    active#(half(X)) -> active#(X) ->
    active#(sqr(s(X))) -> add#(sqr(X),dbl(X))
    active#(half(X)) -> active#(X) -> active#(sqr(s(X))) -> sqr#(X)
    active#(half(X)) -> active#(X) -> active#(sqr(s(X))) -> dbl#(X)
    active#(half(X)) -> active#(X) ->
    active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N)))
    active#(half(X)) -> active#(X) ->
    active#(terms(N)) -> recip#(sqr(N))
    active#(half(X)) -> active#(X) -> active#(terms(N)) -> sqr#(N)
    active#(half(X)) -> active#(X) ->
    active#(terms(N)) -> terms#(s(N))
    active#(half(X)) -> active#(X) ->
    active#(terms(N)) -> s#(N)
    active#(first(s(X),cons(Y,Z))) -> first#(X,Z) ->
    first#(ok(X1),ok(X2)) -> first#(X1,X2)
    active#(first(s(X),cons(Y,Z))) -> first#(X,Z) ->
    first#(X1,mark(X2)) -> first#(X1,X2)
    active#(first(s(X),cons(Y,Z))) -> first#(X,Z) ->
    first#(mark(X1),X2) -> first#(X1,X2)
    active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) ->
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z)) ->
    cons#(mark(X1),X2) -> cons#(X1,X2)
    active#(first(X1,X2)) -> first#(active(X1),X2) ->
    first#(ok(X1),ok(X2)) -> first#(X1,X2)
    active#(first(X1,X2)) -> first#(active(X1),X2) ->
    first#(X1,mark(X2)) -> first#(X1,X2)
    active#(first(X1,X2)) -> first#(active(X1),X2) ->
    first#(mark(X1),X2) -> first#(X1,X2)
    active#(first(X1,X2)) -> first#(X1,active(X2)) ->
    first#(ok(X1),ok(X2)) -> first#(X1,X2)
    active#(first(X1,X2)) -> first#(X1,active(X2)) ->
    first#(X1,mark(X2)) -> first#(X1,X2)
    active#(first(X1,X2)) -> first#(X1,active(X2)) ->
    first#(mark(X1),X2) -> first#(X1,X2)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(half(X)) -> half#(active(X))
    active#(first(X1,X2)) -> active#(X2) ->
    active#(half(X)) -> active#(X)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(first(X1,X2)) -> first#(X1,active(X2))
    active#(first(X1,X2)) -> active#(X2) ->
    active#(first(X1,X2)) -> active#(X2)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(first(X1,X2)) -> first#(active(X1),X2)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(first(X1,X2)) -> active#(X1)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(dbl(X)) -> dbl#(active(X))
    active#(first(X1,X2)) -> active#(X2) ->
    active#(dbl(X)) -> active#(X)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(add(X1,X2)) -> add#(X1,active(X2))
    active#(first(X1,X2)) -> active#(X2) ->
    active#(add(X1,X2)) -> active#(X2)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(add(X1,X2)) -> add#(active(X1),X2)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(add(X1,X2)) -> active#(X1)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(s(X)) -> s#(active(X))
    active#(first(X1,X2)) -> active#(X2) ->
    active#(s(X)) -> active#(X)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(sqr(X)) -> sqr#(active(X))
    active#(first(X1,X2)) -> active#(X2) ->
    active#(sqr(X)) -> active#(X)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(recip(X)) -> recip#(active(X))
    active#(first(X1,X2)) -> active#(X2) ->
    active#(recip(X)) -> active#(X)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(cons(X1,X2)) -> active#(X1)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(terms(X)) -> terms#(active(X))
    active#(first(X1,X2)) -> active#(X2) ->
    active#(terms(X)) -> active#(X)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(half(s(s(X)))) -> s#(half(X))
    active#(first(X1,X2)) -> active#(X2) ->
    active#(half(s(s(X)))) -> half#(X)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z))
    active#(first(X1,X2)) -> active#(X2) ->
    active#(first(s(X),cons(Y,Z))) -> first#(X,Z)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(add(s(X),Y)) -> s#(add(X,Y))
    active#(first(X1,X2)) -> active#(X2) ->
    active#(add(s(X),Y)) -> add#(X,Y)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(dbl(s(X))) -> s#(s(dbl(X)))
    active#(first(X1,X2)) -> active#(X2) ->
    active#(dbl(s(X))) -> s#(dbl(X))
    active#(first(X1,X2)) -> active#(X2) ->
    active#(dbl(s(X))) -> dbl#(X)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X)))
    active#(first(X1,X2)) -> active#(X2) ->
    active#(sqr(s(X))) -> add#(sqr(X),dbl(X))
    active#(first(X1,X2)) -> active#(X2) ->
    active#(sqr(s(X))) -> sqr#(X)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(sqr(s(X))) -> dbl#(X)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N)))
    active#(first(X1,X2)) -> active#(X2) ->
    active#(terms(N)) -> recip#(sqr(N))
    active#(first(X1,X2)) -> active#(X2) ->
    active#(terms(N)) -> sqr#(N)
    active#(first(X1,X2)) -> active#(X2) ->
    active#(terms(N)) -> terms#(s(N))
    active#(first(X1,X2)) -> active#(X2) ->
    active#(terms(N)) -> s#(N)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(half(X)) -> half#(active(X))
    active#(first(X1,X2)) -> active#(X1) ->
    active#(half(X)) -> active#(X)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(first(X1,X2)) -> first#(X1,active(X2))
    active#(first(X1,X2)) -> active#(X1) ->
    active#(first(X1,X2)) -> active#(X2)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(first(X1,X2)) -> first#(active(X1),X2)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(first(X1,X2)) -> active#(X1)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(dbl(X)) -> dbl#(active(X))
    active#(first(X1,X2)) -> active#(X1) ->
    active#(dbl(X)) -> active#(X)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(add(X1,X2)) -> add#(X1,active(X2))
    active#(first(X1,X2)) -> active#(X1) ->
    active#(add(X1,X2)) -> active#(X2)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(add(X1,X2)) -> add#(active(X1),X2)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(add(X1,X2)) -> active#(X1)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> s#(active(X))
    active#(first(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> active#(X)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(sqr(X)) -> sqr#(active(X))
    active#(first(X1,X2)) -> active#(X1) ->
    active#(sqr(X)) -> active#(X)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(recip(X)) -> recip#(active(X))
    active#(first(X1,X2)) -> active#(X1) ->
    active#(recip(X)) -> active#(X)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(cons(X1,X2)) -> active#(X1)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(terms(X)) -> terms#(active(X))
    active#(first(X1,X2)) -> active#(X1) ->
    active#(terms(X)) -> active#(X)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(half(s(s(X)))) -> s#(half(X))
    active#(first(X1,X2)) -> active#(X1) ->
    active#(half(s(s(X)))) -> half#(X)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z))
    active#(first(X1,X2)) -> active#(X1) ->
    active#(first(s(X),cons(Y,Z))) -> first#(X,Z)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(add(s(X),Y)) -> s#(add(X,Y))
    active#(first(X1,X2)) -> active#(X1) ->
    active#(add(s(X),Y)) -> add#(X,Y)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(dbl(s(X))) -> s#(s(dbl(X)))
    active#(first(X1,X2)) -> active#(X1) ->
    active#(dbl(s(X))) -> s#(dbl(X))
    active#(first(X1,X2)) -> active#(X1) ->
    active#(dbl(s(X))) -> dbl#(X)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X)))
    active#(first(X1,X2)) -> active#(X1) ->
    active#(sqr(s(X))) -> add#(sqr(X),dbl(X))
    active#(first(X1,X2)) -> active#(X1) ->
    active#(sqr(s(X))) -> sqr#(X)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(sqr(s(X))) -> dbl#(X)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N)))
    active#(first(X1,X2)) -> active#(X1) ->
    active#(terms(N)) -> recip#(sqr(N))
    active#(first(X1,X2)) -> active#(X1) ->
    active#(terms(N)) -> sqr#(N)
    active#(first(X1,X2)) -> active#(X1) ->
    active#(terms(N)) -> terms#(s(N))
    active#(first(X1,X2)) -> active#(X1) ->
    active#(terms(N)) -> s#(N)
    active#(add(s(X),Y)) -> add#(X,Y) ->
    add#(ok(X1),ok(X2)) -> add#(X1,X2)
    active#(add(s(X),Y)) -> add#(X,Y) ->
    add#(X1,mark(X2)) -> add#(X1,X2)
    active#(add(s(X),Y)) -> add#(X,Y) ->
    add#(mark(X1),X2) -> add#(X1,X2)
    active#(add(s(X),Y)) -> s#(add(X,Y)) -> s#(ok(X)) -> s#(X)
    active#(add(s(X),Y)) -> s#(add(X,Y)) ->
    s#(mark(X)) -> s#(X)
    active#(add(X1,X2)) -> add#(active(X1),X2) ->
    add#(ok(X1),ok(X2)) -> add#(X1,X2)
    active#(add(X1,X2)) -> add#(active(X1),X2) ->
    add#(X1,mark(X2)) -> add#(X1,X2)
    active#(add(X1,X2)) -> add#(active(X1),X2) ->
    add#(mark(X1),X2) -> add#(X1,X2)
    active#(add(X1,X2)) -> add#(X1,active(X2)) ->
    add#(ok(X1),ok(X2)) -> add#(X1,X2)
    active#(add(X1,X2)) -> add#(X1,active(X2)) ->
    add#(X1,mark(X2)) -> add#(X1,X2)
    active#(add(X1,X2)) -> add#(X1,active(X2)) ->
    add#(mark(X1),X2) -> add#(X1,X2)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(half(X)) -> half#(active(X))
    active#(add(X1,X2)) -> active#(X2) ->
    active#(half(X)) -> active#(X)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(first(X1,X2)) -> first#(X1,active(X2))
    active#(add(X1,X2)) -> active#(X2) ->
    active#(first(X1,X2)) -> active#(X2)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(first(X1,X2)) -> first#(active(X1),X2)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(first(X1,X2)) -> active#(X1)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(dbl(X)) -> dbl#(active(X))
    active#(add(X1,X2)) -> active#(X2) ->
    active#(dbl(X)) -> active#(X)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(add(X1,X2)) -> add#(X1,active(X2))
    active#(add(X1,X2)) -> active#(X2) ->
    active#(add(X1,X2)) -> active#(X2)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(add(X1,X2)) -> add#(active(X1),X2)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(add(X1,X2)) -> active#(X1)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(s(X)) -> s#(active(X))
    active#(add(X1,X2)) -> active#(X2) ->
    active#(s(X)) -> active#(X)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(sqr(X)) -> sqr#(active(X))
    active#(add(X1,X2)) -> active#(X2) ->
    active#(sqr(X)) -> active#(X)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(recip(X)) -> recip#(active(X))
    active#(add(X1,X2)) -> active#(X2) ->
    active#(recip(X)) -> active#(X)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(cons(X1,X2)) -> active#(X1)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(terms(X)) -> terms#(active(X))
    active#(add(X1,X2)) -> active#(X2) ->
    active#(terms(X)) -> active#(X)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(half(s(s(X)))) -> s#(half(X))
    active#(add(X1,X2)) -> active#(X2) ->
    active#(half(s(s(X)))) -> half#(X)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z))
    active#(add(X1,X2)) -> active#(X2) ->
    active#(first(s(X),cons(Y,Z))) -> first#(X,Z)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(add(s(X),Y)) -> s#(add(X,Y))
    active#(add(X1,X2)) -> active#(X2) ->
    active#(add(s(X),Y)) -> add#(X,Y)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(dbl(s(X))) -> s#(s(dbl(X)))
    active#(add(X1,X2)) -> active#(X2) ->
    active#(dbl(s(X))) -> s#(dbl(X))
    active#(add(X1,X2)) -> active#(X2) ->
    active#(dbl(s(X))) -> dbl#(X)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X)))
    active#(add(X1,X2)) -> active#(X2) ->
    active#(sqr(s(X))) -> add#(sqr(X),dbl(X))
    active#(add(X1,X2)) -> active#(X2) ->
    active#(sqr(s(X))) -> sqr#(X)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(sqr(s(X))) -> dbl#(X)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N)))
    active#(add(X1,X2)) -> active#(X2) ->
    active#(terms(N)) -> recip#(sqr(N))
    active#(add(X1,X2)) -> active#(X2) ->
    active#(terms(N)) -> sqr#(N)
    active#(add(X1,X2)) -> active#(X2) ->
    active#(terms(N)) -> terms#(s(N))
    active#(add(X1,X2)) -> active#(X2) ->
    active#(terms(N)) -> s#(N)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(half(X)) -> half#(active(X))
    active#(add(X1,X2)) -> active#(X1) ->
    active#(half(X)) -> active#(X)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(first(X1,X2)) -> first#(X1,active(X2))
    active#(add(X1,X2)) -> active#(X1) ->
    active#(first(X1,X2)) -> active#(X2)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(first(X1,X2)) -> first#(active(X1),X2)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(first(X1,X2)) -> active#(X1)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(dbl(X)) -> dbl#(active(X))
    active#(add(X1,X2)) -> active#(X1) ->
    active#(dbl(X)) -> active#(X)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(add(X1,X2)) -> add#(X1,active(X2))
    active#(add(X1,X2)) -> active#(X1) ->
    active#(add(X1,X2)) -> active#(X2)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(add(X1,X2)) -> add#(active(X1),X2)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(add(X1,X2)) -> active#(X1)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> s#(active(X))
    active#(add(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> active#(X)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(sqr(X)) -> sqr#(active(X))
    active#(add(X1,X2)) -> active#(X1) ->
    active#(sqr(X)) -> active#(X)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(recip(X)) -> recip#(active(X))
    active#(add(X1,X2)) -> active#(X1) ->
    active#(recip(X)) -> active#(X)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(cons(X1,X2)) -> active#(X1)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(terms(X)) -> terms#(active(X))
    active#(add(X1,X2)) -> active#(X1) ->
    active#(terms(X)) -> active#(X)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(half(s(s(X)))) -> s#(half(X))
    active#(add(X1,X2)) -> active#(X1) ->
    active#(half(s(s(X)))) -> half#(X)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z))
    active#(add(X1,X2)) -> active#(X1) ->
    active#(first(s(X),cons(Y,Z))) -> first#(X,Z)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(add(s(X),Y)) -> s#(add(X,Y))
    active#(add(X1,X2)) -> active#(X1) ->
    active#(add(s(X),Y)) -> add#(X,Y)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(dbl(s(X))) -> s#(s(dbl(X)))
    active#(add(X1,X2)) -> active#(X1) ->
    active#(dbl(s(X))) -> s#(dbl(X))
    active#(add(X1,X2)) -> active#(X1) ->
    active#(dbl(s(X))) -> dbl#(X)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X)))
    active#(add(X1,X2)) -> active#(X1) ->
    active#(sqr(s(X))) -> add#(sqr(X),dbl(X))
    active#(add(X1,X2)) -> active#(X1) ->
    active#(sqr(s(X))) -> sqr#(X)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(sqr(s(X))) -> dbl#(X)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N)))
    active#(add(X1,X2)) -> active#(X1) ->
    active#(terms(N)) -> recip#(sqr(N))
    active#(add(X1,X2)) -> active#(X1) ->
    active#(terms(N)) -> sqr#(N)
    active#(add(X1,X2)) -> active#(X1) ->
    active#(terms(N)) -> terms#(s(N))
    active#(add(X1,X2)) -> active#(X1) -> active#(terms(N)) -> s#(N)
    active#(dbl(s(X))) -> dbl#(X) -> dbl#(ok(X)) -> dbl#(X)
    active#(dbl(s(X))) -> dbl#(X) -> dbl#(mark(X)) -> dbl#(X)
    active#(dbl(s(X))) -> s#(dbl(X)) -> s#(ok(X)) -> s#(X)
    active#(dbl(s(X))) -> s#(dbl(X)) -> s#(mark(X)) -> s#(X)
    active#(dbl(s(X))) -> s#(s(dbl(X))) -> s#(ok(X)) -> s#(X)
    active#(dbl(s(X))) -> s#(s(dbl(X))) -> s#(mark(X)) -> s#(X)
    active#(dbl(X)) -> dbl#(active(X)) -> dbl#(ok(X)) -> dbl#(X)
    active#(dbl(X)) -> dbl#(active(X)) -> dbl#(mark(X)) -> dbl#(X)
    active#(dbl(X)) -> active#(X) ->
    active#(half(X)) -> half#(active(X))
    active#(dbl(X)) -> active#(X) -> active#(half(X)) -> active#(X)
    active#(dbl(X)) -> active#(X) ->
    active#(first(X1,X2)) -> first#(X1,active(X2))
    active#(dbl(X)) -> active#(X) ->
    active#(first(X1,X2)) -> active#(X2)
    active#(dbl(X)) -> active#(X) ->
    active#(first(X1,X2)) -> first#(active(X1),X2)
    active#(dbl(X)) -> active#(X) ->
    active#(first(X1,X2)) -> active#(X1)
    active#(dbl(X)) -> active#(X) ->
    active#(dbl(X)) -> dbl#(active(X))
    active#(dbl(X)) -> active#(X) -> active#(dbl(X)) -> active#(X)
    active#(dbl(X)) -> active#(X) ->
    active#(add(X1,X2)) -> add#(X1,active(X2))
    active#(dbl(X)) -> active#(X) ->
    active#(add(X1,X2)) -> active#(X2)
    active#(dbl(X)) -> active#(X) ->
    active#(add(X1,X2)) -> add#(active(X1),X2)
    active#(dbl(X)) -> active#(X) ->
    active#(add(X1,X2)) -> active#(X1)
    active#(dbl(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
    active#(dbl(X)) -> active#(X) -> active#(s(X)) -> active#(X)
    active#(dbl(X)) -> active#(X) ->
    active#(sqr(X)) -> sqr#(active(X))
    active#(dbl(X)) -> active#(X) -> active#(sqr(X)) -> active#(X)
    active#(dbl(X)) -> active#(X) ->
    active#(recip(X)) -> recip#(active(X))
    active#(dbl(X)) -> active#(X) -> active#(recip(X)) -> active#(X)
    active#(dbl(X)) -> active#(X) ->
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(dbl(X)) -> active#(X) ->
    active#(cons(X1,X2)) -> active#(X1)
    active#(dbl(X)) -> active#(X) ->
    active#(terms(X)) -> terms#(active(X))
    active#(dbl(X)) -> active#(X) -> active#(terms(X)) -> active#(X)
    active#(dbl(X)) -> active#(X) ->
    active#(half(s(s(X)))) -> s#(half(X))
    active#(dbl(X)) -> active#(X) ->
    active#(half(s(s(X)))) -> half#(X)
    active#(dbl(X)) -> active#(X) ->
    active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z))
    active#(dbl(X)) -> active#(X) ->
    active#(first(s(X),cons(Y,Z))) -> first#(X,Z)
    active#(dbl(X)) -> active#(X) ->
    active#(add(s(X),Y)) -> s#(add(X,Y))
    active#(dbl(X)) -> active#(X) -> active#(add(s(X),Y)) -> add#(X,Y)
    active#(dbl(X)) -> active#(X) ->
    active#(dbl(s(X))) -> s#(s(dbl(X)))
    active#(dbl(X)) -> active#(X) -> active#(dbl(s(X))) -> s#(dbl(X))
    active#(dbl(X)) -> active#(X) -> active#(dbl(s(X))) -> dbl#(X)
    active#(dbl(X)) -> active#(X) ->
    active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X)))
    active#(dbl(X)) -> active#(X) ->
    active#(sqr(s(X))) -> add#(sqr(X),dbl(X))
    active#(dbl(X)) -> active#(X) -> active#(sqr(s(X))) -> sqr#(X)
    active#(dbl(X)) -> active#(X) -> active#(sqr(s(X))) -> dbl#(X)
    active#(dbl(X)) -> active#(X) ->
    active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N)))
    active#(dbl(X)) -> active#(X) ->
    active#(terms(N)) -> recip#(sqr(N))
    active#(dbl(X)) -> active#(X) -> active#(terms(N)) -> sqr#(N)
    active#(dbl(X)) -> active#(X) -> active#(terms(N)) -> terms#(s(N))
    active#(dbl(X)) -> active#(X) ->
    active#(terms(N)) -> s#(N)
    active#(cons(X1,X2)) -> cons#(active(X1),X2) ->
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    active#(cons(X1,X2)) -> cons#(active(X1),X2) ->
    cons#(mark(X1),X2) -> cons#(X1,X2)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(half(X)) -> half#(active(X))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(half(X)) -> active#(X)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(first(X1,X2)) -> first#(X1,active(X2))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(first(X1,X2)) -> active#(X2)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(first(X1,X2)) -> first#(active(X1),X2)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(first(X1,X2)) -> active#(X1)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(dbl(X)) -> dbl#(active(X))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(dbl(X)) -> active#(X)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(add(X1,X2)) -> add#(X1,active(X2))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(add(X1,X2)) -> active#(X2)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(add(X1,X2)) -> add#(active(X1),X2)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(add(X1,X2)) -> active#(X1)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> s#(active(X))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(s(X)) -> active#(X)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(sqr(X)) -> sqr#(active(X))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(sqr(X)) -> active#(X)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(recip(X)) -> recip#(active(X))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(recip(X)) -> active#(X)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(cons(X1,X2)) -> active#(X1)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(terms(X)) -> terms#(active(X))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(terms(X)) -> active#(X)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(half(s(s(X)))) -> s#(half(X))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(half(s(s(X)))) -> half#(X)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(first(s(X),cons(Y,Z))) -> first#(X,Z)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(add(s(X),Y)) -> s#(add(X,Y))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(add(s(X),Y)) -> add#(X,Y)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(dbl(s(X))) -> s#(s(dbl(X)))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(dbl(s(X))) -> s#(dbl(X))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(dbl(s(X))) -> dbl#(X)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X)))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(sqr(s(X))) -> add#(sqr(X),dbl(X))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(sqr(s(X))) -> sqr#(X)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(sqr(s(X))) -> dbl#(X)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N)))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(terms(N)) -> recip#(sqr(N))
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(terms(N)) -> sqr#(N)
    active#(cons(X1,X2)) -> active#(X1) ->
    active#(terms(N)) -> terms#(s(N))
    active#(cons(X1,X2)) -> active#(X1) -> active#(terms(N)) -> s#(N)
    active#(s(X)) -> s#(active(X)) -> s#(ok(X)) -> s#(X)
    active#(s(X)) -> s#(active(X)) -> s#(mark(X)) -> s#(X)
    active#(s(X)) -> active#(X) -> active#(half(X)) -> half#(active(X))
    active#(s(X)) -> active#(X) -> active#(half(X)) -> active#(X)
    active#(s(X)) -> active#(X) ->
    active#(first(X1,X2)) -> first#(X1,active(X2))
    active#(s(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X2)
    active#(s(X)) -> active#(X) ->
    active#(first(X1,X2)) -> first#(active(X1),X2)
    active#(s(X)) -> active#(X) -> active#(first(X1,X2)) -> active#(X1)
    active#(s(X)) -> active#(X) -> active#(dbl(X)) -> dbl#(active(X))
    active#(s(X)) -> active#(X) -> active#(dbl(X)) -> active#(X)
    active#(s(X)) -> active#(X) ->
    active#(add(X1,X2)) -> add#(X1,active(X2))
    active#(s(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X2)
    active#(s(X)) -> active#(X) ->
    active#(add(X1,X2)) -> add#(active(X1),X2)
    active#(s(X)) -> active#(X) -> active#(add(X1,X2)) -> active#(X1)
    active#(s(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
    active#(s(X)) -> active#(X) -> active#(s(X)) -> active#(X)
    active#(s(X)) -> active#(X) -> active#(sqr(X)) -> sqr#(active(X))
    active#(s(X)) -> active#(X) -> active#(sqr(X)) -> active#(X)
    active#(s(X)) -> active#(X) ->
    active#(recip(X)) -> recip#(active(X))
    active#(s(X)) -> active#(X) -> active#(recip(X)) -> active#(X)
    active#(s(X)) -> active#(X) ->
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(s(X)) -> active#(X) -> active#(cons(X1,X2)) -> active#(X1)
    active#(s(X)) -> active#(X) ->
    active#(terms(X)) -> terms#(active(X))
    active#(s(X)) -> active#(X) -> active#(terms(X)) -> active#(X)
    active#(s(X)) -> active#(X) -> active#(half(s(s(X)))) -> s#(half(X))
    active#(s(X)) -> active#(X) -> active#(half(s(s(X)))) -> half#(X)
    active#(s(X)) -> active#(X) ->
    active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z))
    active#(s(X)) -> active#(X) ->
    active#(first(s(X),cons(Y,Z))) -> first#(X,Z)
    active#(s(X)) -> active#(X) -> active#(add(s(X),Y)) -> s#(add(X,Y))
    active#(s(X)) -> active#(X) -> active#(add(s(X),Y)) -> add#(X,Y)
    active#(s(X)) -> active#(X) -> active#(dbl(s(X))) -> s#(s(dbl(X)))
    active#(s(X)) -> active#(X) -> active#(dbl(s(X))) -> s#(dbl(X))
    active#(s(X)) -> active#(X) -> active#(dbl(s(X))) -> dbl#(X)
    active#(s(X)) -> active#(X) ->
    active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X)))
    active#(s(X)) -> active#(X) ->
    active#(sqr(s(X))) -> add#(sqr(X),dbl(X))
    active#(s(X)) -> active#(X) -> active#(sqr(s(X))) -> sqr#(X)
    active#(s(X)) -> active#(X) -> active#(sqr(s(X))) -> dbl#(X)
    active#(s(X)) -> active#(X) ->
    active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N)))
    active#(s(X)) -> active#(X) -> active#(terms(N)) -> recip#(sqr(N))
    active#(s(X)) -> active#(X) -> active#(terms(N)) -> sqr#(N)
    active#(s(X)) -> active#(X) -> active#(terms(N)) -> terms#(s(N))
    active#(s(X)) -> active#(X) -> active#(terms(N)) -> s#(N)
    active#(recip(X)) -> recip#(active(X)) ->
    recip#(ok(X)) -> recip#(X)
    active#(recip(X)) -> recip#(active(X)) ->
    recip#(mark(X)) -> recip#(X)
    active#(recip(X)) -> active#(X) ->
    active#(half(X)) -> half#(active(X))
    active#(recip(X)) -> active#(X) ->
    active#(half(X)) -> active#(X)
    active#(recip(X)) -> active#(X) ->
    active#(first(X1,X2)) -> first#(X1,active(X2))
    active#(recip(X)) -> active#(X) ->
    active#(first(X1,X2)) -> active#(X2)
    active#(recip(X)) -> active#(X) ->
    active#(first(X1,X2)) -> first#(active(X1),X2)
    active#(recip(X)) -> active#(X) ->
    active#(first(X1,X2)) -> active#(X1)
    active#(recip(X)) -> active#(X) ->
    active#(dbl(X)) -> dbl#(active(X))
    active#(recip(X)) -> active#(X) -> active#(dbl(X)) -> active#(X)
    active#(recip(X)) -> active#(X) ->
    active#(add(X1,X2)) -> add#(X1,active(X2))
    active#(recip(X)) -> active#(X) ->
    active#(add(X1,X2)) -> active#(X2)
    active#(recip(X)) -> active#(X) ->
    active#(add(X1,X2)) -> add#(active(X1),X2)
    active#(recip(X)) -> active#(X) ->
    active#(add(X1,X2)) -> active#(X1)
    active#(recip(X)) -> active#(X) ->
    active#(s(X)) -> s#(active(X))
    active#(recip(X)) -> active#(X) -> active#(s(X)) -> active#(X)
    active#(recip(X)) -> active#(X) ->
    active#(sqr(X)) -> sqr#(active(X))
    active#(recip(X)) -> active#(X) -> active#(sqr(X)) -> active#(X)
    active#(recip(X)) -> active#(X) ->
    active#(recip(X)) -> recip#(active(X))
    active#(recip(X)) -> active#(X) ->
    active#(recip(X)) -> active#(X)
    active#(recip(X)) -> active#(X) ->
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(recip(X)) -> active#(X) ->
    active#(cons(X1,X2)) -> active#(X1)
    active#(recip(X)) -> active#(X) ->
    active#(terms(X)) -> terms#(active(X))
    active#(recip(X)) -> active#(X) ->
    active#(terms(X)) -> active#(X)
    active#(recip(X)) -> active#(X) ->
    active#(half(s(s(X)))) -> s#(half(X))
    active#(recip(X)) -> active#(X) ->
    active#(half(s(s(X)))) -> half#(X)
    active#(recip(X)) -> active#(X) ->
    active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z))
    active#(recip(X)) -> active#(X) ->
    active#(first(s(X),cons(Y,Z))) -> first#(X,Z)
    active#(recip(X)) -> active#(X) ->
    active#(add(s(X),Y)) -> s#(add(X,Y))
    active#(recip(X)) -> active#(X) ->
    active#(add(s(X),Y)) -> add#(X,Y)
    active#(recip(X)) -> active#(X) ->
    active#(dbl(s(X))) -> s#(s(dbl(X)))
    active#(recip(X)) -> active#(X) ->
    active#(dbl(s(X))) -> s#(dbl(X))
    active#(recip(X)) -> active#(X) -> active#(dbl(s(X))) -> dbl#(X)
    active#(recip(X)) -> active#(X) ->
    active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X)))
    active#(recip(X)) -> active#(X) ->
    active#(sqr(s(X))) -> add#(sqr(X),dbl(X))
    active#(recip(X)) -> active#(X) -> active#(sqr(s(X))) -> sqr#(X)
    active#(recip(X)) -> active#(X) -> active#(sqr(s(X))) -> dbl#(X)
    active#(recip(X)) -> active#(X) ->
    active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N)))
    active#(recip(X)) -> active#(X) ->
    active#(terms(N)) -> recip#(sqr(N))
    active#(recip(X)) -> active#(X) -> active#(terms(N)) -> sqr#(N)
    active#(recip(X)) -> active#(X) ->
    active#(terms(N)) -> terms#(s(N))
    active#(recip(X)) -> active#(X) ->
    active#(terms(N)) -> s#(N)
    active#(sqr(s(X))) -> add#(sqr(X),dbl(X)) ->
    add#(ok(X1),ok(X2)) -> add#(X1,X2)
    active#(sqr(s(X))) -> add#(sqr(X),dbl(X)) ->
    add#(X1,mark(X2)) -> add#(X1,X2)
    active#(sqr(s(X))) -> add#(sqr(X),dbl(X)) ->
    add#(mark(X1),X2) -> add#(X1,X2)
    active#(sqr(s(X))) -> dbl#(X) -> dbl#(ok(X)) -> dbl#(X)
    active#(sqr(s(X))) -> dbl#(X) -> dbl#(mark(X)) -> dbl#(X)
    active#(sqr(s(X))) -> sqr#(X) -> sqr#(ok(X)) -> sqr#(X)
    active#(sqr(s(X))) -> sqr#(X) ->
    sqr#(mark(X)) -> sqr#(X)
    active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X))) ->
    s#(ok(X)) -> s#(X)
    active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X))) ->
    s#(mark(X)) -> s#(X)
    active#(sqr(X)) -> sqr#(active(X)) -> sqr#(ok(X)) -> sqr#(X)
    active#(sqr(X)) -> sqr#(active(X)) -> sqr#(mark(X)) -> sqr#(X)
    active#(sqr(X)) -> active#(X) ->
    active#(half(X)) -> half#(active(X))
    active#(sqr(X)) -> active#(X) -> active#(half(X)) -> active#(X)
    active#(sqr(X)) -> active#(X) ->
    active#(first(X1,X2)) -> first#(X1,active(X2))
    active#(sqr(X)) -> active#(X) ->
    active#(first(X1,X2)) -> active#(X2)
    active#(sqr(X)) -> active#(X) ->
    active#(first(X1,X2)) -> first#(active(X1),X2)
    active#(sqr(X)) -> active#(X) ->
    active#(first(X1,X2)) -> active#(X1)
    active#(sqr(X)) -> active#(X) ->
    active#(dbl(X)) -> dbl#(active(X))
    active#(sqr(X)) -> active#(X) -> active#(dbl(X)) -> active#(X)
    active#(sqr(X)) -> active#(X) ->
    active#(add(X1,X2)) -> add#(X1,active(X2))
    active#(sqr(X)) -> active#(X) ->
    active#(add(X1,X2)) -> active#(X2)
    active#(sqr(X)) -> active#(X) ->
    active#(add(X1,X2)) -> add#(active(X1),X2)
    active#(sqr(X)) -> active#(X) ->
    active#(add(X1,X2)) -> active#(X1)
    active#(sqr(X)) -> active#(X) -> active#(s(X)) -> s#(active(X))
    active#(sqr(X)) -> active#(X) -> active#(s(X)) -> active#(X)
    active#(sqr(X)) -> active#(X) ->
    active#(sqr(X)) -> sqr#(active(X))
    active#(sqr(X)) -> active#(X) -> active#(sqr(X)) -> active#(X)
    active#(sqr(X)) -> active#(X) ->
    active#(recip(X)) -> recip#(active(X))
    active#(sqr(X)) -> active#(X) -> active#(recip(X)) -> active#(X)
    active#(sqr(X)) -> active#(X) ->
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(sqr(X)) -> active#(X) ->
    active#(cons(X1,X2)) -> active#(X1)
    active#(sqr(X)) -> active#(X) ->
    active#(terms(X)) -> terms#(active(X))
    active#(sqr(X)) -> active#(X) -> active#(terms(X)) -> active#(X)
    active#(sqr(X)) -> active#(X) ->
    active#(half(s(s(X)))) -> s#(half(X))
    active#(sqr(X)) -> active#(X) ->
    active#(half(s(s(X)))) -> half#(X)
    active#(sqr(X)) -> active#(X) ->
    active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z))
    active#(sqr(X)) -> active#(X) ->
    active#(first(s(X),cons(Y,Z))) -> first#(X,Z)
    active#(sqr(X)) -> active#(X) ->
    active#(add(s(X),Y)) -> s#(add(X,Y))
    active#(sqr(X)) -> active#(X) -> active#(add(s(X),Y)) -> add#(X,Y)
    active#(sqr(X)) -> active#(X) ->
    active#(dbl(s(X))) -> s#(s(dbl(X)))
    active#(sqr(X)) -> active#(X) -> active#(dbl(s(X))) -> s#(dbl(X))
    active#(sqr(X)) -> active#(X) -> active#(dbl(s(X))) -> dbl#(X)
    active#(sqr(X)) -> active#(X) ->
    active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X)))
    active#(sqr(X)) -> active#(X) ->
    active#(sqr(s(X))) -> add#(sqr(X),dbl(X))
    active#(sqr(X)) -> active#(X) -> active#(sqr(s(X))) -> sqr#(X)
    active#(sqr(X)) -> active#(X) -> active#(sqr(s(X))) -> dbl#(X)
    active#(sqr(X)) -> active#(X) ->
    active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N)))
    active#(sqr(X)) -> active#(X) ->
    active#(terms(N)) -> recip#(sqr(N))
    active#(sqr(X)) -> active#(X) -> active#(terms(N)) -> sqr#(N)
    active#(sqr(X)) -> active#(X) -> active#(terms(N)) -> terms#(s(N))
    active#(sqr(X)) -> active#(X) ->
    active#(terms(N)) -> s#(N)
    active#(terms(X)) -> terms#(active(X)) ->
    terms#(ok(X)) -> terms#(X)
    active#(terms(X)) -> terms#(active(X)) ->
    terms#(mark(X)) -> terms#(X)
    active#(terms(X)) -> active#(X) ->
    active#(half(X)) -> half#(active(X))
    active#(terms(X)) -> active#(X) ->
    active#(half(X)) -> active#(X)
    active#(terms(X)) -> active#(X) ->
    active#(first(X1,X2)) -> first#(X1,active(X2))
    active#(terms(X)) -> active#(X) ->
    active#(first(X1,X2)) -> active#(X2)
    active#(terms(X)) -> active#(X) ->
    active#(first(X1,X2)) -> first#(active(X1),X2)
    active#(terms(X)) -> active#(X) ->
    active#(first(X1,X2)) -> active#(X1)
    active#(terms(X)) -> active#(X) ->
    active#(dbl(X)) -> dbl#(active(X))
    active#(terms(X)) -> active#(X) -> active#(dbl(X)) -> active#(X)
    active#(terms(X)) -> active#(X) ->
    active#(add(X1,X2)) -> add#(X1,active(X2))
    active#(terms(X)) -> active#(X) ->
    active#(add(X1,X2)) -> active#(X2)
    active#(terms(X)) -> active#(X) ->
    active#(add(X1,X2)) -> add#(active(X1),X2)
    active#(terms(X)) -> active#(X) ->
    active#(add(X1,X2)) -> active#(X1)
    active#(terms(X)) -> active#(X) ->
    active#(s(X)) -> s#(active(X))
    active#(terms(X)) -> active#(X) -> active#(s(X)) -> active#(X)
    active#(terms(X)) -> active#(X) ->
    active#(sqr(X)) -> sqr#(active(X))
    active#(terms(X)) -> active#(X) -> active#(sqr(X)) -> active#(X)
    active#(terms(X)) -> active#(X) ->
    active#(recip(X)) -> recip#(active(X))
    active#(terms(X)) -> active#(X) ->
    active#(recip(X)) -> active#(X)
    active#(terms(X)) -> active#(X) ->
    active#(cons(X1,X2)) -> cons#(active(X1),X2)
    active#(terms(X)) -> active#(X) ->
    active#(cons(X1,X2)) -> active#(X1)
    active#(terms(X)) -> active#(X) ->
    active#(terms(X)) -> terms#(active(X))
    active#(terms(X)) -> active#(X) ->
    active#(terms(X)) -> active#(X)
    active#(terms(X)) -> active#(X) ->
    active#(half(s(s(X)))) -> s#(half(X))
    active#(terms(X)) -> active#(X) ->
    active#(half(s(s(X)))) -> half#(X)
    active#(terms(X)) -> active#(X) ->
    active#(first(s(X),cons(Y,Z))) -> cons#(Y,first(X,Z))
    active#(terms(X)) -> active#(X) ->
    active#(first(s(X),cons(Y,Z))) -> first#(X,Z)
    active#(terms(X)) -> active#(X) ->
    active#(add(s(X),Y)) -> s#(add(X,Y))
    active#(terms(X)) -> active#(X) ->
    active#(add(s(X),Y)) -> add#(X,Y)
    active#(terms(X)) -> active#(X) ->
    active#(dbl(s(X))) -> s#(s(dbl(X)))
    active#(terms(X)) -> active#(X) ->
    active#(dbl(s(X))) -> s#(dbl(X))
    active#(terms(X)) -> active#(X) -> active#(dbl(s(X))) -> dbl#(X)
    active#(terms(X)) -> active#(X) ->
    active#(sqr(s(X))) -> s#(add(sqr(X),dbl(X)))
    active#(terms(X)) -> active#(X) ->
    active#(sqr(s(X))) -> add#(sqr(X),dbl(X))
    active#(terms(X)) -> active#(X) -> active#(sqr(s(X))) -> sqr#(X)
    active#(terms(X)) -> active#(X) -> active#(sqr(s(X))) -> dbl#(X)
    active#(terms(X)) -> active#(X) ->
    active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N)))
    active#(terms(X)) -> active#(X) ->
    active#(terms(N)) -> recip#(sqr(N))
    active#(terms(X)) -> active#(X) -> active#(terms(N)) -> sqr#(N)
    active#(terms(X)) -> active#(X) ->
    active#(terms(N)) -> terms#(s(N))
    active#(terms(X)) -> active#(X) ->
    active#(terms(N)) -> s#(N)
    active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N))) ->
    cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    active#(terms(N)) -> cons#(recip(sqr(N)),terms(s(N))) ->
    cons#(mark(X1),X2) -> cons#(X1,X2)
    active#(terms(N)) -> recip#(sqr(N)) ->
    recip#(ok(X)) -> recip#(X)
    active#(terms(N)) -> recip#(sqr(N)) -> recip#(mark(X)) -> recip#(X)
    active#(terms(N)) -> sqr#(N) -> sqr#(ok(X)) -> sqr#(X)
    active#(terms(N)) -> sqr#(N) -> sqr#(mark(X)) -> sqr#(X)
    active#(terms(N)) -> terms#(s(N)) ->
    terms#(ok(X)) -> terms#(X)
    active#(terms(N)) -> terms#(s(N)) -> terms#(mark(X)) -> terms#(X)
    active#(terms(N)) -> s#(N) -> s#(ok(X)) -> s#(X)
    active#(terms(N)) -> s#(N) -> s#(mark(X)) -> s#(X)
   SCC Processor:
    #sccs: 12
    #rules: 45
    #arcs: 892/7225
    DPs:
     top#(ok(X)) -> top#(active(X))
     top#(mark(X)) -> top#(proper(X))
    TRS:
     active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
     active(sqr(0())) -> mark(0())
     active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
     active(dbl(0())) -> mark(0())
     active(dbl(s(X))) -> mark(s(s(dbl(X))))
     active(add(0(),X)) -> mark(X)
     active(add(s(X),Y)) -> mark(s(add(X,Y)))
     active(first(0(),X)) -> mark(nil())
     active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
     active(half(0())) -> mark(0())
     active(half(s(0()))) -> mark(0())
     active(half(s(s(X)))) -> mark(s(half(X)))
     active(half(dbl(X))) -> mark(X)
     active(terms(X)) -> terms(active(X))
     active(cons(X1,X2)) -> cons(active(X1),X2)
     active(recip(X)) -> recip(active(X))
     active(sqr(X)) -> sqr(active(X))
     active(s(X)) -> s(active(X))
     active(add(X1,X2)) -> add(active(X1),X2)
     active(add(X1,X2)) -> add(X1,active(X2))
     active(dbl(X)) -> dbl(active(X))
     active(first(X1,X2)) -> first(active(X1),X2)
     active(first(X1,X2)) -> first(X1,active(X2))
     active(half(X)) -> half(active(X))
     terms(mark(X)) -> mark(terms(X))
     cons(mark(X1),X2) -> mark(cons(X1,X2))
     recip(mark(X)) -> mark(recip(X))
     sqr(mark(X)) -> mark(sqr(X))
     s(mark(X)) -> mark(s(X))
     add(mark(X1),X2) -> mark(add(X1,X2))
     add(X1,mark(X2)) -> mark(add(X1,X2))
     dbl(mark(X)) -> mark(dbl(X))
     first(mark(X1),X2) -> mark(first(X1,X2))
     first(X1,mark(X2)) -> mark(first(X1,X2))
     half(mark(X)) -> mark(half(X))
     proper(terms(X)) -> terms(proper(X))
     proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
     proper(recip(X)) -> recip(proper(X))
     proper(sqr(X)) -> sqr(proper(X))
     proper(s(X)) -> s(proper(X))
     proper(0()) -> ok(0())
     proper(add(X1,X2)) -> add(proper(X1),proper(X2))
     proper(dbl(X)) -> dbl(proper(X))
     proper(first(X1,X2)) -> first(proper(X1),proper(X2))
     proper(nil()) -> ok(nil())
     proper(half(X)) -> half(proper(X))
     terms(ok(X)) -> ok(terms(X))
     cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
     recip(ok(X)) -> ok(recip(X))
     sqr(ok(X)) -> ok(sqr(X))
     s(ok(X)) -> ok(s(X))
     add(ok(X1),ok(X2)) -> ok(add(X1,X2))
     dbl(ok(X)) -> ok(dbl(X))
     first(ok(X1),ok(X2)) -> ok(first(X1,X2))
     half(ok(X)) -> ok(half(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Usable Rule Processor:
     DPs:
      top#(ok(X)) -> top#(active(X))
      top#(mark(X)) -> top#(proper(X))
     TRS:
      active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
      active(sqr(0())) -> mark(0())
      active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
      active(dbl(0())) -> mark(0())
      active(dbl(s(X))) -> mark(s(s(dbl(X))))
      active(add(0(),X)) -> mark(X)
      active(add(s(X),Y)) -> mark(s(add(X,Y)))
      active(first(0(),X)) -> mark(nil())
      active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
      active(half(0())) -> mark(0())
      active(half(s(0()))) -> mark(0())
      active(half(s(s(X)))) -> mark(s(half(X)))
      active(half(dbl(X))) -> mark(X)
      active(terms(X)) -> terms(active(X))
      active(cons(X1,X2)) -> cons(active(X1),X2)
      active(recip(X)) -> recip(active(X))
      active(sqr(X)) -> sqr(active(X))
      active(s(X)) -> s(active(X))
      active(add(X1,X2)) -> add(active(X1),X2)
      active(add(X1,X2)) -> add(X1,active(X2))
      active(dbl(X)) -> dbl(active(X))
      active(first(X1,X2)) -> first(active(X1),X2)
      active(first(X1,X2)) -> first(X1,active(X2))
      active(half(X)) -> half(active(X))
      s(mark(X)) -> mark(s(X))
      s(ok(X)) -> ok(s(X))
      terms(mark(X)) -> mark(terms(X))
      terms(ok(X)) -> ok(terms(X))
      sqr(mark(X)) -> mark(sqr(X))
      sqr(ok(X)) -> ok(sqr(X))
      recip(mark(X)) -> mark(recip(X))
      recip(ok(X)) -> ok(recip(X))
      cons(mark(X1),X2) -> mark(cons(X1,X2))
      cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
      dbl(mark(X)) -> mark(dbl(X))
      dbl(ok(X)) -> ok(dbl(X))
      add(mark(X1),X2) -> mark(add(X1,X2))
      add(X1,mark(X2)) -> mark(add(X1,X2))
      add(ok(X1),ok(X2)) -> ok(add(X1,X2))
      first(mark(X1),X2) -> mark(first(X1,X2))
      first(X1,mark(X2)) -> mark(first(X1,X2))
      first(ok(X1),ok(X2)) -> ok(first(X1,X2))
      half(mark(X)) -> mark(half(X))
      half(ok(X)) -> ok(half(X))
      proper(terms(X)) -> terms(proper(X))
      proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
      proper(recip(X)) -> recip(proper(X))
      proper(sqr(X)) -> sqr(proper(X))
      proper(s(X)) -> s(proper(X))
      proper(0()) -> ok(0())
      proper(add(X1,X2)) -> add(proper(X1),proper(X2))
      proper(dbl(X)) -> dbl(proper(X))
      proper(first(X1,X2)) -> first(proper(X1),proper(X2))
      proper(nil()) -> ok(nil())
      proper(half(X)) -> half(proper(X))
     LPO Processor:
      argument filtering:
       pi(terms) = [0]
       pi(active) = 0
       pi(sqr) = [0]
       pi(recip) = 0
       pi(s) = [0]
       pi(cons) = 0
       pi(mark) = [0]
       pi(0) = []
       pi(dbl) = [0]
       pi(add) = [0,1]
       pi(first) = [0,1]
       pi(nil) = []
       pi(half) = [0]
       pi(proper) = 0
       pi(ok) = 0
       pi(top#) = 0
      usable rules:
       active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
       active(sqr(0())) -> mark(0())
       active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
       active(dbl(0())) -> mark(0())
       active(dbl(s(X))) -> mark(s(s(dbl(X))))
       active(add(0(),X)) -> mark(X)
       active(add(s(X),Y)) -> mark(s(add(X,Y)))
       active(first(0(),X)) -> mark(nil())
       active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
       active(half(0())) -> mark(0())
       active(half(s(0()))) -> mark(0())
       active(half(s(s(X)))) -> mark(s(half(X)))
       active(half(dbl(X))) -> mark(X)
       active(terms(X)) -> terms(active(X))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(recip(X)) -> recip(active(X))
       active(sqr(X)) -> sqr(active(X))
       active(s(X)) -> s(active(X))
       active(add(X1,X2)) -> add(active(X1),X2)
       active(add(X1,X2)) -> add(X1,active(X2))
       active(dbl(X)) -> dbl(active(X))
       active(first(X1,X2)) -> first(active(X1),X2)
       active(first(X1,X2)) -> first(X1,active(X2))
       active(half(X)) -> half(active(X))
       s(mark(X)) -> mark(s(X))
       s(ok(X)) -> ok(s(X))
       terms(mark(X)) -> mark(terms(X))
       terms(ok(X)) -> ok(terms(X))
       sqr(mark(X)) -> mark(sqr(X))
       sqr(ok(X)) -> ok(sqr(X))
       recip(mark(X)) -> mark(recip(X))
       recip(ok(X)) -> ok(recip(X))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       dbl(mark(X)) -> mark(dbl(X))
       dbl(ok(X)) -> ok(dbl(X))
       add(mark(X1),X2) -> mark(add(X1,X2))
       add(X1,mark(X2)) -> mark(add(X1,X2))
       add(ok(X1),ok(X2)) -> ok(add(X1,X2))
       first(mark(X1),X2) -> mark(first(X1,X2))
       first(X1,mark(X2)) -> mark(first(X1,X2))
       first(ok(X1),ok(X2)) -> ok(first(X1,X2))
       half(mark(X)) -> mark(half(X))
       half(ok(X)) -> ok(half(X))
       proper(terms(X)) -> terms(proper(X))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(recip(X)) -> recip(proper(X))
       proper(sqr(X)) -> sqr(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(0()) -> ok(0())
       proper(add(X1,X2)) -> add(proper(X1),proper(X2))
       proper(dbl(X)) -> dbl(proper(X))
       proper(first(X1,X2)) -> first(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(half(X)) -> half(proper(X))
      precedence:
       terms > sqr > add > dbl > first > s > half > top# ~ ok ~ proper ~ 
       nil ~ 0 ~ mark ~ cons ~ recip ~ active
      problem:
       DPs:
        top#(ok(X)) -> top#(active(X))
       TRS:
        active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
        active(sqr(0())) -> mark(0())
        active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
        active(dbl(0())) -> mark(0())
        active(dbl(s(X))) -> mark(s(s(dbl(X))))
        active(add(0(),X)) -> mark(X)
        active(add(s(X),Y)) -> mark(s(add(X,Y)))
        active(first(0(),X)) -> mark(nil())
        active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
        active(half(0())) -> mark(0())
        active(half(s(0()))) -> mark(0())
        active(half(s(s(X)))) -> mark(s(half(X)))
        active(half(dbl(X))) -> mark(X)
        active(terms(X)) -> terms(active(X))
        active(cons(X1,X2)) -> cons(active(X1),X2)
        active(recip(X)) -> recip(active(X))
        active(sqr(X)) -> sqr(active(X))
        active(s(X)) -> s(active(X))
        active(add(X1,X2)) -> add(active(X1),X2)
        active(add(X1,X2)) -> add(X1,active(X2))
        active(dbl(X)) -> dbl(active(X))
        active(first(X1,X2)) -> first(active(X1),X2)
        active(first(X1,X2)) -> first(X1,active(X2))
        active(half(X)) -> half(active(X))
        s(mark(X)) -> mark(s(X))
        s(ok(X)) -> ok(s(X))
        terms(mark(X)) -> mark(terms(X))
        terms(ok(X)) -> ok(terms(X))
        sqr(mark(X)) -> mark(sqr(X))
        sqr(ok(X)) -> ok(sqr(X))
        recip(mark(X)) -> mark(recip(X))
        recip(ok(X)) -> ok(recip(X))
        cons(mark(X1),X2) -> mark(cons(X1,X2))
        cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
        dbl(mark(X)) -> mark(dbl(X))
        dbl(ok(X)) -> ok(dbl(X))
        add(mark(X1),X2) -> mark(add(X1,X2))
        add(X1,mark(X2)) -> mark(add(X1,X2))
        add(ok(X1),ok(X2)) -> ok(add(X1,X2))
        first(mark(X1),X2) -> mark(first(X1,X2))
        first(X1,mark(X2)) -> mark(first(X1,X2))
        first(ok(X1),ok(X2)) -> ok(first(X1,X2))
        half(mark(X)) -> mark(half(X))
        half(ok(X)) -> ok(half(X))
        proper(terms(X)) -> terms(proper(X))
        proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
        proper(recip(X)) -> recip(proper(X))
        proper(sqr(X)) -> sqr(proper(X))
        proper(s(X)) -> s(proper(X))
        proper(0()) -> ok(0())
        proper(add(X1,X2)) -> add(proper(X1),proper(X2))
        proper(dbl(X)) -> dbl(proper(X))
        proper(first(X1,X2)) -> first(proper(X1),proper(X2))
        proper(nil()) -> ok(nil())
        proper(half(X)) -> half(proper(X))
      Restore Modifier:
       DPs:
        top#(ok(X)) -> top#(active(X))
       TRS:
        active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
        active(sqr(0())) -> mark(0())
        active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
        active(dbl(0())) -> mark(0())
        active(dbl(s(X))) -> mark(s(s(dbl(X))))
        active(add(0(),X)) -> mark(X)
        active(add(s(X),Y)) -> mark(s(add(X,Y)))
        active(first(0(),X)) -> mark(nil())
        active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
        active(half(0())) -> mark(0())
        active(half(s(0()))) -> mark(0())
        active(half(s(s(X)))) -> mark(s(half(X)))
        active(half(dbl(X))) -> mark(X)
        active(terms(X)) -> terms(active(X))
        active(cons(X1,X2)) -> cons(active(X1),X2)
        active(recip(X)) -> recip(active(X))
        active(sqr(X)) -> sqr(active(X))
        active(s(X)) -> s(active(X))
        active(add(X1,X2)) -> add(active(X1),X2)
        active(add(X1,X2)) -> add(X1,active(X2))
        active(dbl(X)) -> dbl(active(X))
        active(first(X1,X2)) -> first(active(X1),X2)
        active(first(X1,X2)) -> first(X1,active(X2))
        active(half(X)) -> half(active(X))
        terms(mark(X)) -> mark(terms(X))
        cons(mark(X1),X2) -> mark(cons(X1,X2))
        recip(mark(X)) -> mark(recip(X))
        sqr(mark(X)) -> mark(sqr(X))
        s(mark(X)) -> mark(s(X))
        add(mark(X1),X2) -> mark(add(X1,X2))
        add(X1,mark(X2)) -> mark(add(X1,X2))
        dbl(mark(X)) -> mark(dbl(X))
        first(mark(X1),X2) -> mark(first(X1,X2))
        first(X1,mark(X2)) -> mark(first(X1,X2))
        half(mark(X)) -> mark(half(X))
        proper(terms(X)) -> terms(proper(X))
        proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
        proper(recip(X)) -> recip(proper(X))
        proper(sqr(X)) -> sqr(proper(X))
        proper(s(X)) -> s(proper(X))
        proper(0()) -> ok(0())
        proper(add(X1,X2)) -> add(proper(X1),proper(X2))
        proper(dbl(X)) -> dbl(proper(X))
        proper(first(X1,X2)) -> first(proper(X1),proper(X2))
        proper(nil()) -> ok(nil())
        proper(half(X)) -> half(proper(X))
        terms(ok(X)) -> ok(terms(X))
        cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
        recip(ok(X)) -> ok(recip(X))
        sqr(ok(X)) -> ok(sqr(X))
        s(ok(X)) -> ok(s(X))
        add(ok(X1),ok(X2)) -> ok(add(X1,X2))
        dbl(ok(X)) -> ok(dbl(X))
        first(ok(X1),ok(X2)) -> ok(first(X1,X2))
        half(ok(X)) -> ok(half(X))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
       Usable Rule Processor:
        DPs:
         top#(ok(X)) -> top#(active(X))
        TRS:
         active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
         active(sqr(0())) -> mark(0())
         active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
         active(dbl(0())) -> mark(0())
         active(dbl(s(X))) -> mark(s(s(dbl(X))))
         active(add(0(),X)) -> mark(X)
         active(add(s(X),Y)) -> mark(s(add(X,Y)))
         active(first(0(),X)) -> mark(nil())
         active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
         active(half(0())) -> mark(0())
         active(half(s(0()))) -> mark(0())
         active(half(s(s(X)))) -> mark(s(half(X)))
         active(half(dbl(X))) -> mark(X)
         active(terms(X)) -> terms(active(X))
         active(cons(X1,X2)) -> cons(active(X1),X2)
         active(recip(X)) -> recip(active(X))
         active(sqr(X)) -> sqr(active(X))
         active(s(X)) -> s(active(X))
         active(add(X1,X2)) -> add(active(X1),X2)
         active(add(X1,X2)) -> add(X1,active(X2))
         active(dbl(X)) -> dbl(active(X))
         active(first(X1,X2)) -> first(active(X1),X2)
         active(first(X1,X2)) -> first(X1,active(X2))
         active(half(X)) -> half(active(X))
         s(mark(X)) -> mark(s(X))
         s(ok(X)) -> ok(s(X))
         terms(mark(X)) -> mark(terms(X))
         terms(ok(X)) -> ok(terms(X))
         sqr(mark(X)) -> mark(sqr(X))
         sqr(ok(X)) -> ok(sqr(X))
         recip(mark(X)) -> mark(recip(X))
         recip(ok(X)) -> ok(recip(X))
         cons(mark(X1),X2) -> mark(cons(X1,X2))
         cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
         dbl(mark(X)) -> mark(dbl(X))
         dbl(ok(X)) -> ok(dbl(X))
         add(mark(X1),X2) -> mark(add(X1,X2))
         add(X1,mark(X2)) -> mark(add(X1,X2))
         add(ok(X1),ok(X2)) -> ok(add(X1,X2))
         first(mark(X1),X2) -> mark(first(X1,X2))
         first(X1,mark(X2)) -> mark(first(X1,X2))
         first(ok(X1),ok(X2)) -> ok(first(X1,X2))
         half(mark(X)) -> mark(half(X))
         half(ok(X)) -> ok(half(X))
        KBO Processor:
         argument filtering:
          pi(terms) = 0
          pi(active) = [0]
          pi(sqr) = 0
          pi(recip) = 0
          pi(s) = 0
          pi(cons) = 1
          pi(mark) = 0
          pi(0) = []
          pi(dbl) = 0
          pi(add) = 1
          pi(first) = 1
          pi(nil) = []
          pi(half) = 0
          pi(ok) = [0]
          pi(top#) = 0
         usable rules:
          active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
          active(sqr(0())) -> mark(0())
          active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
          active(dbl(0())) -> mark(0())
          active(dbl(s(X))) -> mark(s(s(dbl(X))))
          active(add(0(),X)) -> mark(X)
          active(add(s(X),Y)) -> mark(s(add(X,Y)))
          active(first(0(),X)) -> mark(nil())
          active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
          active(half(0())) -> mark(0())
          active(half(s(0()))) -> mark(0())
          active(half(s(s(X)))) -> mark(s(half(X)))
          active(half(dbl(X))) -> mark(X)
          active(terms(X)) -> terms(active(X))
          active(cons(X1,X2)) -> cons(active(X1),X2)
          active(recip(X)) -> recip(active(X))
          active(sqr(X)) -> sqr(active(X))
          active(s(X)) -> s(active(X))
          active(add(X1,X2)) -> add(active(X1),X2)
          active(add(X1,X2)) -> add(X1,active(X2))
          active(dbl(X)) -> dbl(active(X))
          active(first(X1,X2)) -> first(active(X1),X2)
          active(first(X1,X2)) -> first(X1,active(X2))
          active(half(X)) -> half(active(X))
          s(mark(X)) -> mark(s(X))
          s(ok(X)) -> ok(s(X))
          terms(mark(X)) -> mark(terms(X))
          terms(ok(X)) -> ok(terms(X))
          sqr(mark(X)) -> mark(sqr(X))
          sqr(ok(X)) -> ok(sqr(X))
          recip(mark(X)) -> mark(recip(X))
          recip(ok(X)) -> ok(recip(X))
          cons(mark(X1),X2) -> mark(cons(X1,X2))
          cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
          dbl(mark(X)) -> mark(dbl(X))
          dbl(ok(X)) -> ok(dbl(X))
          add(mark(X1),X2) -> mark(add(X1,X2))
          add(X1,mark(X2)) -> mark(add(X1,X2))
          add(ok(X1),ok(X2)) -> ok(add(X1,X2))
          first(mark(X1),X2) -> mark(first(X1,X2))
          first(X1,mark(X2)) -> mark(first(X1,X2))
          first(ok(X1),ok(X2)) -> ok(first(X1,X2))
          half(mark(X)) -> mark(half(X))
          half(ok(X)) -> ok(half(X))
         weight function:
          w0 = 1
          w(top#) = w(ok) = w(nil) = w(0) = w(active) = 1
          w(half) = w(first) = w(add) = w(dbl) = w(mark) = w(cons) = w(
          s) = w(recip) = w(sqr) = w(terms) = 0
         precedence:
          ok > top# ~ half ~ nil ~ first ~ add ~ dbl ~ 0 ~ mark ~ cons ~ 
          s ~ recip ~ sqr ~ active ~ terms
         problem:
          DPs:
           
          TRS:
           active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
           active(sqr(0())) -> mark(0())
           active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
           active(dbl(0())) -> mark(0())
           active(dbl(s(X))) -> mark(s(s(dbl(X))))
           active(add(0(),X)) -> mark(X)
           active(add(s(X),Y)) -> mark(s(add(X,Y)))
           active(first(0(),X)) -> mark(nil())
           active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
           active(half(0())) -> mark(0())
           active(half(s(0()))) -> mark(0())
           active(half(s(s(X)))) -> mark(s(half(X)))
           active(half(dbl(X))) -> mark(X)
           active(terms(X)) -> terms(active(X))
           active(cons(X1,X2)) -> cons(active(X1),X2)
           active(recip(X)) -> recip(active(X))
           active(sqr(X)) -> sqr(active(X))
           active(s(X)) -> s(active(X))
           active(add(X1,X2)) -> add(active(X1),X2)
           active(add(X1,X2)) -> add(X1,active(X2))
           active(dbl(X)) -> dbl(active(X))
           active(first(X1,X2)) -> first(active(X1),X2)
           active(first(X1,X2)) -> first(X1,active(X2))
           active(half(X)) -> half(active(X))
           s(mark(X)) -> mark(s(X))
           s(ok(X)) -> ok(s(X))
           terms(mark(X)) -> mark(terms(X))
           terms(ok(X)) -> ok(terms(X))
           sqr(mark(X)) -> mark(sqr(X))
           sqr(ok(X)) -> ok(sqr(X))
           recip(mark(X)) -> mark(recip(X))
           recip(ok(X)) -> ok(recip(X))
           cons(mark(X1),X2) -> mark(cons(X1,X2))
           cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
           dbl(mark(X)) -> mark(dbl(X))
           dbl(ok(X)) -> ok(dbl(X))
           add(mark(X1),X2) -> mark(add(X1,X2))
           add(X1,mark(X2)) -> mark(add(X1,X2))
           add(ok(X1),ok(X2)) -> ok(add(X1,X2))
           first(mark(X1),X2) -> mark(first(X1,X2))
           first(X1,mark(X2)) -> mark(first(X1,X2))
           first(ok(X1),ok(X2)) -> ok(first(X1,X2))
           half(mark(X)) -> mark(half(X))
           half(ok(X)) -> ok(half(X))
         Qed
    
    DPs:
     active#(terms(X)) -> active#(X)
     active#(cons(X1,X2)) -> active#(X1)
     active#(recip(X)) -> active#(X)
     active#(sqr(X)) -> active#(X)
     active#(s(X)) -> active#(X)
     active#(add(X1,X2)) -> active#(X1)
     active#(add(X1,X2)) -> active#(X2)
     active#(dbl(X)) -> active#(X)
     active#(first(X1,X2)) -> active#(X1)
     active#(first(X1,X2)) -> active#(X2)
     active#(half(X)) -> active#(X)
    TRS:
     active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
     active(sqr(0())) -> mark(0())
     active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
     active(dbl(0())) -> mark(0())
     active(dbl(s(X))) -> mark(s(s(dbl(X))))
     active(add(0(),X)) -> mark(X)
     active(add(s(X),Y)) -> mark(s(add(X,Y)))
     active(first(0(),X)) -> mark(nil())
     active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
     active(half(0())) -> mark(0())
     active(half(s(0()))) -> mark(0())
     active(half(s(s(X)))) -> mark(s(half(X)))
     active(half(dbl(X))) -> mark(X)
     active(terms(X)) -> terms(active(X))
     active(cons(X1,X2)) -> cons(active(X1),X2)
     active(recip(X)) -> recip(active(X))
     active(sqr(X)) -> sqr(active(X))
     active(s(X)) -> s(active(X))
     active(add(X1,X2)) -> add(active(X1),X2)
     active(add(X1,X2)) -> add(X1,active(X2))
     active(dbl(X)) -> dbl(active(X))
     active(first(X1,X2)) -> first(active(X1),X2)
     active(first(X1,X2)) -> first(X1,active(X2))
     active(half(X)) -> half(active(X))
     terms(mark(X)) -> mark(terms(X))
     cons(mark(X1),X2) -> mark(cons(X1,X2))
     recip(mark(X)) -> mark(recip(X))
     sqr(mark(X)) -> mark(sqr(X))
     s(mark(X)) -> mark(s(X))
     add(mark(X1),X2) -> mark(add(X1,X2))
     add(X1,mark(X2)) -> mark(add(X1,X2))
     dbl(mark(X)) -> mark(dbl(X))
     first(mark(X1),X2) -> mark(first(X1,X2))
     first(X1,mark(X2)) -> mark(first(X1,X2))
     half(mark(X)) -> mark(half(X))
     proper(terms(X)) -> terms(proper(X))
     proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
     proper(recip(X)) -> recip(proper(X))
     proper(sqr(X)) -> sqr(proper(X))
     proper(s(X)) -> s(proper(X))
     proper(0()) -> ok(0())
     proper(add(X1,X2)) -> add(proper(X1),proper(X2))
     proper(dbl(X)) -> dbl(proper(X))
     proper(first(X1,X2)) -> first(proper(X1),proper(X2))
     proper(nil()) -> ok(nil())
     proper(half(X)) -> half(proper(X))
     terms(ok(X)) -> ok(terms(X))
     cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
     recip(ok(X)) -> ok(recip(X))
     sqr(ok(X)) -> ok(sqr(X))
     s(ok(X)) -> ok(s(X))
     add(ok(X1),ok(X2)) -> ok(add(X1,X2))
     dbl(ok(X)) -> ok(dbl(X))
     first(ok(X1),ok(X2)) -> ok(first(X1,X2))
     half(ok(X)) -> ok(half(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(active#) = 0
     problem:
      DPs:
       
      TRS:
       active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
       active(sqr(0())) -> mark(0())
       active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
       active(dbl(0())) -> mark(0())
       active(dbl(s(X))) -> mark(s(s(dbl(X))))
       active(add(0(),X)) -> mark(X)
       active(add(s(X),Y)) -> mark(s(add(X,Y)))
       active(first(0(),X)) -> mark(nil())
       active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
       active(half(0())) -> mark(0())
       active(half(s(0()))) -> mark(0())
       active(half(s(s(X)))) -> mark(s(half(X)))
       active(half(dbl(X))) -> mark(X)
       active(terms(X)) -> terms(active(X))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(recip(X)) -> recip(active(X))
       active(sqr(X)) -> sqr(active(X))
       active(s(X)) -> s(active(X))
       active(add(X1,X2)) -> add(active(X1),X2)
       active(add(X1,X2)) -> add(X1,active(X2))
       active(dbl(X)) -> dbl(active(X))
       active(first(X1,X2)) -> first(active(X1),X2)
       active(first(X1,X2)) -> first(X1,active(X2))
       active(half(X)) -> half(active(X))
       terms(mark(X)) -> mark(terms(X))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       recip(mark(X)) -> mark(recip(X))
       sqr(mark(X)) -> mark(sqr(X))
       s(mark(X)) -> mark(s(X))
       add(mark(X1),X2) -> mark(add(X1,X2))
       add(X1,mark(X2)) -> mark(add(X1,X2))
       dbl(mark(X)) -> mark(dbl(X))
       first(mark(X1),X2) -> mark(first(X1,X2))
       first(X1,mark(X2)) -> mark(first(X1,X2))
       half(mark(X)) -> mark(half(X))
       proper(terms(X)) -> terms(proper(X))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(recip(X)) -> recip(proper(X))
       proper(sqr(X)) -> sqr(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(0()) -> ok(0())
       proper(add(X1,X2)) -> add(proper(X1),proper(X2))
       proper(dbl(X)) -> dbl(proper(X))
       proper(first(X1,X2)) -> first(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(half(X)) -> half(proper(X))
       terms(ok(X)) -> ok(terms(X))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       recip(ok(X)) -> ok(recip(X))
       sqr(ok(X)) -> ok(sqr(X))
       s(ok(X)) -> ok(s(X))
       add(ok(X1),ok(X2)) -> ok(add(X1,X2))
       dbl(ok(X)) -> ok(dbl(X))
       first(ok(X1),ok(X2)) -> ok(first(X1,X2))
       half(ok(X)) -> ok(half(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     proper#(terms(X)) -> proper#(X)
     proper#(cons(X1,X2)) -> proper#(X2)
     proper#(cons(X1,X2)) -> proper#(X1)
     proper#(recip(X)) -> proper#(X)
     proper#(sqr(X)) -> proper#(X)
     proper#(s(X)) -> proper#(X)
     proper#(add(X1,X2)) -> proper#(X2)
     proper#(add(X1,X2)) -> proper#(X1)
     proper#(dbl(X)) -> proper#(X)
     proper#(first(X1,X2)) -> proper#(X2)
     proper#(first(X1,X2)) -> proper#(X1)
     proper#(half(X)) -> proper#(X)
    TRS:
     active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
     active(sqr(0())) -> mark(0())
     active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
     active(dbl(0())) -> mark(0())
     active(dbl(s(X))) -> mark(s(s(dbl(X))))
     active(add(0(),X)) -> mark(X)
     active(add(s(X),Y)) -> mark(s(add(X,Y)))
     active(first(0(),X)) -> mark(nil())
     active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
     active(half(0())) -> mark(0())
     active(half(s(0()))) -> mark(0())
     active(half(s(s(X)))) -> mark(s(half(X)))
     active(half(dbl(X))) -> mark(X)
     active(terms(X)) -> terms(active(X))
     active(cons(X1,X2)) -> cons(active(X1),X2)
     active(recip(X)) -> recip(active(X))
     active(sqr(X)) -> sqr(active(X))
     active(s(X)) -> s(active(X))
     active(add(X1,X2)) -> add(active(X1),X2)
     active(add(X1,X2)) -> add(X1,active(X2))
     active(dbl(X)) -> dbl(active(X))
     active(first(X1,X2)) -> first(active(X1),X2)
     active(first(X1,X2)) -> first(X1,active(X2))
     active(half(X)) -> half(active(X))
     terms(mark(X)) -> mark(terms(X))
     cons(mark(X1),X2) -> mark(cons(X1,X2))
     recip(mark(X)) -> mark(recip(X))
     sqr(mark(X)) -> mark(sqr(X))
     s(mark(X)) -> mark(s(X))
     add(mark(X1),X2) -> mark(add(X1,X2))
     add(X1,mark(X2)) -> mark(add(X1,X2))
     dbl(mark(X)) -> mark(dbl(X))
     first(mark(X1),X2) -> mark(first(X1,X2))
     first(X1,mark(X2)) -> mark(first(X1,X2))
     half(mark(X)) -> mark(half(X))
     proper(terms(X)) -> terms(proper(X))
     proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
     proper(recip(X)) -> recip(proper(X))
     proper(sqr(X)) -> sqr(proper(X))
     proper(s(X)) -> s(proper(X))
     proper(0()) -> ok(0())
     proper(add(X1,X2)) -> add(proper(X1),proper(X2))
     proper(dbl(X)) -> dbl(proper(X))
     proper(first(X1,X2)) -> first(proper(X1),proper(X2))
     proper(nil()) -> ok(nil())
     proper(half(X)) -> half(proper(X))
     terms(ok(X)) -> ok(terms(X))
     cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
     recip(ok(X)) -> ok(recip(X))
     sqr(ok(X)) -> ok(sqr(X))
     s(ok(X)) -> ok(s(X))
     add(ok(X1),ok(X2)) -> ok(add(X1,X2))
     dbl(ok(X)) -> ok(dbl(X))
     first(ok(X1),ok(X2)) -> ok(first(X1,X2))
     half(ok(X)) -> ok(half(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(proper#) = 0
     problem:
      DPs:
       
      TRS:
       active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
       active(sqr(0())) -> mark(0())
       active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
       active(dbl(0())) -> mark(0())
       active(dbl(s(X))) -> mark(s(s(dbl(X))))
       active(add(0(),X)) -> mark(X)
       active(add(s(X),Y)) -> mark(s(add(X,Y)))
       active(first(0(),X)) -> mark(nil())
       active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
       active(half(0())) -> mark(0())
       active(half(s(0()))) -> mark(0())
       active(half(s(s(X)))) -> mark(s(half(X)))
       active(half(dbl(X))) -> mark(X)
       active(terms(X)) -> terms(active(X))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(recip(X)) -> recip(active(X))
       active(sqr(X)) -> sqr(active(X))
       active(s(X)) -> s(active(X))
       active(add(X1,X2)) -> add(active(X1),X2)
       active(add(X1,X2)) -> add(X1,active(X2))
       active(dbl(X)) -> dbl(active(X))
       active(first(X1,X2)) -> first(active(X1),X2)
       active(first(X1,X2)) -> first(X1,active(X2))
       active(half(X)) -> half(active(X))
       terms(mark(X)) -> mark(terms(X))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       recip(mark(X)) -> mark(recip(X))
       sqr(mark(X)) -> mark(sqr(X))
       s(mark(X)) -> mark(s(X))
       add(mark(X1),X2) -> mark(add(X1,X2))
       add(X1,mark(X2)) -> mark(add(X1,X2))
       dbl(mark(X)) -> mark(dbl(X))
       first(mark(X1),X2) -> mark(first(X1,X2))
       first(X1,mark(X2)) -> mark(first(X1,X2))
       half(mark(X)) -> mark(half(X))
       proper(terms(X)) -> terms(proper(X))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(recip(X)) -> recip(proper(X))
       proper(sqr(X)) -> sqr(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(0()) -> ok(0())
       proper(add(X1,X2)) -> add(proper(X1),proper(X2))
       proper(dbl(X)) -> dbl(proper(X))
       proper(first(X1,X2)) -> first(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(half(X)) -> half(proper(X))
       terms(ok(X)) -> ok(terms(X))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       recip(ok(X)) -> ok(recip(X))
       sqr(ok(X)) -> ok(sqr(X))
       s(ok(X)) -> ok(s(X))
       add(ok(X1),ok(X2)) -> ok(add(X1,X2))
       dbl(ok(X)) -> ok(dbl(X))
       first(ok(X1),ok(X2)) -> ok(first(X1,X2))
       half(ok(X)) -> ok(half(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     half#(mark(X)) -> half#(X)
     half#(ok(X)) -> half#(X)
    TRS:
     active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
     active(sqr(0())) -> mark(0())
     active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
     active(dbl(0())) -> mark(0())
     active(dbl(s(X))) -> mark(s(s(dbl(X))))
     active(add(0(),X)) -> mark(X)
     active(add(s(X),Y)) -> mark(s(add(X,Y)))
     active(first(0(),X)) -> mark(nil())
     active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
     active(half(0())) -> mark(0())
     active(half(s(0()))) -> mark(0())
     active(half(s(s(X)))) -> mark(s(half(X)))
     active(half(dbl(X))) -> mark(X)
     active(terms(X)) -> terms(active(X))
     active(cons(X1,X2)) -> cons(active(X1),X2)
     active(recip(X)) -> recip(active(X))
     active(sqr(X)) -> sqr(active(X))
     active(s(X)) -> s(active(X))
     active(add(X1,X2)) -> add(active(X1),X2)
     active(add(X1,X2)) -> add(X1,active(X2))
     active(dbl(X)) -> dbl(active(X))
     active(first(X1,X2)) -> first(active(X1),X2)
     active(first(X1,X2)) -> first(X1,active(X2))
     active(half(X)) -> half(active(X))
     terms(mark(X)) -> mark(terms(X))
     cons(mark(X1),X2) -> mark(cons(X1,X2))
     recip(mark(X)) -> mark(recip(X))
     sqr(mark(X)) -> mark(sqr(X))
     s(mark(X)) -> mark(s(X))
     add(mark(X1),X2) -> mark(add(X1,X2))
     add(X1,mark(X2)) -> mark(add(X1,X2))
     dbl(mark(X)) -> mark(dbl(X))
     first(mark(X1),X2) -> mark(first(X1,X2))
     first(X1,mark(X2)) -> mark(first(X1,X2))
     half(mark(X)) -> mark(half(X))
     proper(terms(X)) -> terms(proper(X))
     proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
     proper(recip(X)) -> recip(proper(X))
     proper(sqr(X)) -> sqr(proper(X))
     proper(s(X)) -> s(proper(X))
     proper(0()) -> ok(0())
     proper(add(X1,X2)) -> add(proper(X1),proper(X2))
     proper(dbl(X)) -> dbl(proper(X))
     proper(first(X1,X2)) -> first(proper(X1),proper(X2))
     proper(nil()) -> ok(nil())
     proper(half(X)) -> half(proper(X))
     terms(ok(X)) -> ok(terms(X))
     cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
     recip(ok(X)) -> ok(recip(X))
     sqr(ok(X)) -> ok(sqr(X))
     s(ok(X)) -> ok(s(X))
     add(ok(X1),ok(X2)) -> ok(add(X1,X2))
     dbl(ok(X)) -> ok(dbl(X))
     first(ok(X1),ok(X2)) -> ok(first(X1,X2))
     half(ok(X)) -> ok(half(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(half#) = 0
     problem:
      DPs:
       
      TRS:
       active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
       active(sqr(0())) -> mark(0())
       active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
       active(dbl(0())) -> mark(0())
       active(dbl(s(X))) -> mark(s(s(dbl(X))))
       active(add(0(),X)) -> mark(X)
       active(add(s(X),Y)) -> mark(s(add(X,Y)))
       active(first(0(),X)) -> mark(nil())
       active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
       active(half(0())) -> mark(0())
       active(half(s(0()))) -> mark(0())
       active(half(s(s(X)))) -> mark(s(half(X)))
       active(half(dbl(X))) -> mark(X)
       active(terms(X)) -> terms(active(X))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(recip(X)) -> recip(active(X))
       active(sqr(X)) -> sqr(active(X))
       active(s(X)) -> s(active(X))
       active(add(X1,X2)) -> add(active(X1),X2)
       active(add(X1,X2)) -> add(X1,active(X2))
       active(dbl(X)) -> dbl(active(X))
       active(first(X1,X2)) -> first(active(X1),X2)
       active(first(X1,X2)) -> first(X1,active(X2))
       active(half(X)) -> half(active(X))
       terms(mark(X)) -> mark(terms(X))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       recip(mark(X)) -> mark(recip(X))
       sqr(mark(X)) -> mark(sqr(X))
       s(mark(X)) -> mark(s(X))
       add(mark(X1),X2) -> mark(add(X1,X2))
       add(X1,mark(X2)) -> mark(add(X1,X2))
       dbl(mark(X)) -> mark(dbl(X))
       first(mark(X1),X2) -> mark(first(X1,X2))
       first(X1,mark(X2)) -> mark(first(X1,X2))
       half(mark(X)) -> mark(half(X))
       proper(terms(X)) -> terms(proper(X))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(recip(X)) -> recip(proper(X))
       proper(sqr(X)) -> sqr(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(0()) -> ok(0())
       proper(add(X1,X2)) -> add(proper(X1),proper(X2))
       proper(dbl(X)) -> dbl(proper(X))
       proper(first(X1,X2)) -> first(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(half(X)) -> half(proper(X))
       terms(ok(X)) -> ok(terms(X))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       recip(ok(X)) -> ok(recip(X))
       sqr(ok(X)) -> ok(sqr(X))
       s(ok(X)) -> ok(s(X))
       add(ok(X1),ok(X2)) -> ok(add(X1,X2))
       dbl(ok(X)) -> ok(dbl(X))
       first(ok(X1),ok(X2)) -> ok(first(X1,X2))
       half(ok(X)) -> ok(half(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     first#(mark(X1),X2) -> first#(X1,X2)
     first#(X1,mark(X2)) -> first#(X1,X2)
     first#(ok(X1),ok(X2)) -> first#(X1,X2)
    TRS:
     active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
     active(sqr(0())) -> mark(0())
     active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
     active(dbl(0())) -> mark(0())
     active(dbl(s(X))) -> mark(s(s(dbl(X))))
     active(add(0(),X)) -> mark(X)
     active(add(s(X),Y)) -> mark(s(add(X,Y)))
     active(first(0(),X)) -> mark(nil())
     active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
     active(half(0())) -> mark(0())
     active(half(s(0()))) -> mark(0())
     active(half(s(s(X)))) -> mark(s(half(X)))
     active(half(dbl(X))) -> mark(X)
     active(terms(X)) -> terms(active(X))
     active(cons(X1,X2)) -> cons(active(X1),X2)
     active(recip(X)) -> recip(active(X))
     active(sqr(X)) -> sqr(active(X))
     active(s(X)) -> s(active(X))
     active(add(X1,X2)) -> add(active(X1),X2)
     active(add(X1,X2)) -> add(X1,active(X2))
     active(dbl(X)) -> dbl(active(X))
     active(first(X1,X2)) -> first(active(X1),X2)
     active(first(X1,X2)) -> first(X1,active(X2))
     active(half(X)) -> half(active(X))
     terms(mark(X)) -> mark(terms(X))
     cons(mark(X1),X2) -> mark(cons(X1,X2))
     recip(mark(X)) -> mark(recip(X))
     sqr(mark(X)) -> mark(sqr(X))
     s(mark(X)) -> mark(s(X))
     add(mark(X1),X2) -> mark(add(X1,X2))
     add(X1,mark(X2)) -> mark(add(X1,X2))
     dbl(mark(X)) -> mark(dbl(X))
     first(mark(X1),X2) -> mark(first(X1,X2))
     first(X1,mark(X2)) -> mark(first(X1,X2))
     half(mark(X)) -> mark(half(X))
     proper(terms(X)) -> terms(proper(X))
     proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
     proper(recip(X)) -> recip(proper(X))
     proper(sqr(X)) -> sqr(proper(X))
     proper(s(X)) -> s(proper(X))
     proper(0()) -> ok(0())
     proper(add(X1,X2)) -> add(proper(X1),proper(X2))
     proper(dbl(X)) -> dbl(proper(X))
     proper(first(X1,X2)) -> first(proper(X1),proper(X2))
     proper(nil()) -> ok(nil())
     proper(half(X)) -> half(proper(X))
     terms(ok(X)) -> ok(terms(X))
     cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
     recip(ok(X)) -> ok(recip(X))
     sqr(ok(X)) -> ok(sqr(X))
     s(ok(X)) -> ok(s(X))
     add(ok(X1),ok(X2)) -> ok(add(X1,X2))
     dbl(ok(X)) -> ok(dbl(X))
     first(ok(X1),ok(X2)) -> ok(first(X1,X2))
     half(ok(X)) -> ok(half(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(first#) = 0
     problem:
      DPs:
       first#(X1,mark(X2)) -> first#(X1,X2)
      TRS:
       active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
       active(sqr(0())) -> mark(0())
       active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
       active(dbl(0())) -> mark(0())
       active(dbl(s(X))) -> mark(s(s(dbl(X))))
       active(add(0(),X)) -> mark(X)
       active(add(s(X),Y)) -> mark(s(add(X,Y)))
       active(first(0(),X)) -> mark(nil())
       active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
       active(half(0())) -> mark(0())
       active(half(s(0()))) -> mark(0())
       active(half(s(s(X)))) -> mark(s(half(X)))
       active(half(dbl(X))) -> mark(X)
       active(terms(X)) -> terms(active(X))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(recip(X)) -> recip(active(X))
       active(sqr(X)) -> sqr(active(X))
       active(s(X)) -> s(active(X))
       active(add(X1,X2)) -> add(active(X1),X2)
       active(add(X1,X2)) -> add(X1,active(X2))
       active(dbl(X)) -> dbl(active(X))
       active(first(X1,X2)) -> first(active(X1),X2)
       active(first(X1,X2)) -> first(X1,active(X2))
       active(half(X)) -> half(active(X))
       terms(mark(X)) -> mark(terms(X))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       recip(mark(X)) -> mark(recip(X))
       sqr(mark(X)) -> mark(sqr(X))
       s(mark(X)) -> mark(s(X))
       add(mark(X1),X2) -> mark(add(X1,X2))
       add(X1,mark(X2)) -> mark(add(X1,X2))
       dbl(mark(X)) -> mark(dbl(X))
       first(mark(X1),X2) -> mark(first(X1,X2))
       first(X1,mark(X2)) -> mark(first(X1,X2))
       half(mark(X)) -> mark(half(X))
       proper(terms(X)) -> terms(proper(X))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(recip(X)) -> recip(proper(X))
       proper(sqr(X)) -> sqr(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(0()) -> ok(0())
       proper(add(X1,X2)) -> add(proper(X1),proper(X2))
       proper(dbl(X)) -> dbl(proper(X))
       proper(first(X1,X2)) -> first(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(half(X)) -> half(proper(X))
       terms(ok(X)) -> ok(terms(X))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       recip(ok(X)) -> ok(recip(X))
       sqr(ok(X)) -> ok(sqr(X))
       s(ok(X)) -> ok(s(X))
       add(ok(X1),ok(X2)) -> ok(add(X1,X2))
       dbl(ok(X)) -> ok(dbl(X))
       first(ok(X1),ok(X2)) -> ok(first(X1,X2))
       half(ok(X)) -> ok(half(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(first#) = 1
      problem:
       DPs:
        
       TRS:
        active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
        active(sqr(0())) -> mark(0())
        active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
        active(dbl(0())) -> mark(0())
        active(dbl(s(X))) -> mark(s(s(dbl(X))))
        active(add(0(),X)) -> mark(X)
        active(add(s(X),Y)) -> mark(s(add(X,Y)))
        active(first(0(),X)) -> mark(nil())
        active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
        active(half(0())) -> mark(0())
        active(half(s(0()))) -> mark(0())
        active(half(s(s(X)))) -> mark(s(half(X)))
        active(half(dbl(X))) -> mark(X)
        active(terms(X)) -> terms(active(X))
        active(cons(X1,X2)) -> cons(active(X1),X2)
        active(recip(X)) -> recip(active(X))
        active(sqr(X)) -> sqr(active(X))
        active(s(X)) -> s(active(X))
        active(add(X1,X2)) -> add(active(X1),X2)
        active(add(X1,X2)) -> add(X1,active(X2))
        active(dbl(X)) -> dbl(active(X))
        active(first(X1,X2)) -> first(active(X1),X2)
        active(first(X1,X2)) -> first(X1,active(X2))
        active(half(X)) -> half(active(X))
        terms(mark(X)) -> mark(terms(X))
        cons(mark(X1),X2) -> mark(cons(X1,X2))
        recip(mark(X)) -> mark(recip(X))
        sqr(mark(X)) -> mark(sqr(X))
        s(mark(X)) -> mark(s(X))
        add(mark(X1),X2) -> mark(add(X1,X2))
        add(X1,mark(X2)) -> mark(add(X1,X2))
        dbl(mark(X)) -> mark(dbl(X))
        first(mark(X1),X2) -> mark(first(X1,X2))
        first(X1,mark(X2)) -> mark(first(X1,X2))
        half(mark(X)) -> mark(half(X))
        proper(terms(X)) -> terms(proper(X))
        proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
        proper(recip(X)) -> recip(proper(X))
        proper(sqr(X)) -> sqr(proper(X))
        proper(s(X)) -> s(proper(X))
        proper(0()) -> ok(0())
        proper(add(X1,X2)) -> add(proper(X1),proper(X2))
        proper(dbl(X)) -> dbl(proper(X))
        proper(first(X1,X2)) -> first(proper(X1),proper(X2))
        proper(nil()) -> ok(nil())
        proper(half(X)) -> half(proper(X))
        terms(ok(X)) -> ok(terms(X))
        cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
        recip(ok(X)) -> ok(recip(X))
        sqr(ok(X)) -> ok(sqr(X))
        s(ok(X)) -> ok(s(X))
        add(ok(X1),ok(X2)) -> ok(add(X1,X2))
        dbl(ok(X)) -> ok(dbl(X))
        first(ok(X1),ok(X2)) -> ok(first(X1,X2))
        half(ok(X)) -> ok(half(X))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    DPs:
     dbl#(mark(X)) -> dbl#(X)
     dbl#(ok(X)) -> dbl#(X)
    TRS:
     active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
     active(sqr(0())) -> mark(0())
     active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
     active(dbl(0())) -> mark(0())
     active(dbl(s(X))) -> mark(s(s(dbl(X))))
     active(add(0(),X)) -> mark(X)
     active(add(s(X),Y)) -> mark(s(add(X,Y)))
     active(first(0(),X)) -> mark(nil())
     active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
     active(half(0())) -> mark(0())
     active(half(s(0()))) -> mark(0())
     active(half(s(s(X)))) -> mark(s(half(X)))
     active(half(dbl(X))) -> mark(X)
     active(terms(X)) -> terms(active(X))
     active(cons(X1,X2)) -> cons(active(X1),X2)
     active(recip(X)) -> recip(active(X))
     active(sqr(X)) -> sqr(active(X))
     active(s(X)) -> s(active(X))
     active(add(X1,X2)) -> add(active(X1),X2)
     active(add(X1,X2)) -> add(X1,active(X2))
     active(dbl(X)) -> dbl(active(X))
     active(first(X1,X2)) -> first(active(X1),X2)
     active(first(X1,X2)) -> first(X1,active(X2))
     active(half(X)) -> half(active(X))
     terms(mark(X)) -> mark(terms(X))
     cons(mark(X1),X2) -> mark(cons(X1,X2))
     recip(mark(X)) -> mark(recip(X))
     sqr(mark(X)) -> mark(sqr(X))
     s(mark(X)) -> mark(s(X))
     add(mark(X1),X2) -> mark(add(X1,X2))
     add(X1,mark(X2)) -> mark(add(X1,X2))
     dbl(mark(X)) -> mark(dbl(X))
     first(mark(X1),X2) -> mark(first(X1,X2))
     first(X1,mark(X2)) -> mark(first(X1,X2))
     half(mark(X)) -> mark(half(X))
     proper(terms(X)) -> terms(proper(X))
     proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
     proper(recip(X)) -> recip(proper(X))
     proper(sqr(X)) -> sqr(proper(X))
     proper(s(X)) -> s(proper(X))
     proper(0()) -> ok(0())
     proper(add(X1,X2)) -> add(proper(X1),proper(X2))
     proper(dbl(X)) -> dbl(proper(X))
     proper(first(X1,X2)) -> first(proper(X1),proper(X2))
     proper(nil()) -> ok(nil())
     proper(half(X)) -> half(proper(X))
     terms(ok(X)) -> ok(terms(X))
     cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
     recip(ok(X)) -> ok(recip(X))
     sqr(ok(X)) -> ok(sqr(X))
     s(ok(X)) -> ok(s(X))
     add(ok(X1),ok(X2)) -> ok(add(X1,X2))
     dbl(ok(X)) -> ok(dbl(X))
     first(ok(X1),ok(X2)) -> ok(first(X1,X2))
     half(ok(X)) -> ok(half(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(dbl#) = 0
     problem:
      DPs:
       
      TRS:
       active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
       active(sqr(0())) -> mark(0())
       active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
       active(dbl(0())) -> mark(0())
       active(dbl(s(X))) -> mark(s(s(dbl(X))))
       active(add(0(),X)) -> mark(X)
       active(add(s(X),Y)) -> mark(s(add(X,Y)))
       active(first(0(),X)) -> mark(nil())
       active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
       active(half(0())) -> mark(0())
       active(half(s(0()))) -> mark(0())
       active(half(s(s(X)))) -> mark(s(half(X)))
       active(half(dbl(X))) -> mark(X)
       active(terms(X)) -> terms(active(X))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(recip(X)) -> recip(active(X))
       active(sqr(X)) -> sqr(active(X))
       active(s(X)) -> s(active(X))
       active(add(X1,X2)) -> add(active(X1),X2)
       active(add(X1,X2)) -> add(X1,active(X2))
       active(dbl(X)) -> dbl(active(X))
       active(first(X1,X2)) -> first(active(X1),X2)
       active(first(X1,X2)) -> first(X1,active(X2))
       active(half(X)) -> half(active(X))
       terms(mark(X)) -> mark(terms(X))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       recip(mark(X)) -> mark(recip(X))
       sqr(mark(X)) -> mark(sqr(X))
       s(mark(X)) -> mark(s(X))
       add(mark(X1),X2) -> mark(add(X1,X2))
       add(X1,mark(X2)) -> mark(add(X1,X2))
       dbl(mark(X)) -> mark(dbl(X))
       first(mark(X1),X2) -> mark(first(X1,X2))
       first(X1,mark(X2)) -> mark(first(X1,X2))
       half(mark(X)) -> mark(half(X))
       proper(terms(X)) -> terms(proper(X))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(recip(X)) -> recip(proper(X))
       proper(sqr(X)) -> sqr(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(0()) -> ok(0())
       proper(add(X1,X2)) -> add(proper(X1),proper(X2))
       proper(dbl(X)) -> dbl(proper(X))
       proper(first(X1,X2)) -> first(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(half(X)) -> half(proper(X))
       terms(ok(X)) -> ok(terms(X))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       recip(ok(X)) -> ok(recip(X))
       sqr(ok(X)) -> ok(sqr(X))
       s(ok(X)) -> ok(s(X))
       add(ok(X1),ok(X2)) -> ok(add(X1,X2))
       dbl(ok(X)) -> ok(dbl(X))
       first(ok(X1),ok(X2)) -> ok(first(X1,X2))
       half(ok(X)) -> ok(half(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     add#(mark(X1),X2) -> add#(X1,X2)
     add#(X1,mark(X2)) -> add#(X1,X2)
     add#(ok(X1),ok(X2)) -> add#(X1,X2)
    TRS:
     active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
     active(sqr(0())) -> mark(0())
     active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
     active(dbl(0())) -> mark(0())
     active(dbl(s(X))) -> mark(s(s(dbl(X))))
     active(add(0(),X)) -> mark(X)
     active(add(s(X),Y)) -> mark(s(add(X,Y)))
     active(first(0(),X)) -> mark(nil())
     active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
     active(half(0())) -> mark(0())
     active(half(s(0()))) -> mark(0())
     active(half(s(s(X)))) -> mark(s(half(X)))
     active(half(dbl(X))) -> mark(X)
     active(terms(X)) -> terms(active(X))
     active(cons(X1,X2)) -> cons(active(X1),X2)
     active(recip(X)) -> recip(active(X))
     active(sqr(X)) -> sqr(active(X))
     active(s(X)) -> s(active(X))
     active(add(X1,X2)) -> add(active(X1),X2)
     active(add(X1,X2)) -> add(X1,active(X2))
     active(dbl(X)) -> dbl(active(X))
     active(first(X1,X2)) -> first(active(X1),X2)
     active(first(X1,X2)) -> first(X1,active(X2))
     active(half(X)) -> half(active(X))
     terms(mark(X)) -> mark(terms(X))
     cons(mark(X1),X2) -> mark(cons(X1,X2))
     recip(mark(X)) -> mark(recip(X))
     sqr(mark(X)) -> mark(sqr(X))
     s(mark(X)) -> mark(s(X))
     add(mark(X1),X2) -> mark(add(X1,X2))
     add(X1,mark(X2)) -> mark(add(X1,X2))
     dbl(mark(X)) -> mark(dbl(X))
     first(mark(X1),X2) -> mark(first(X1,X2))
     first(X1,mark(X2)) -> mark(first(X1,X2))
     half(mark(X)) -> mark(half(X))
     proper(terms(X)) -> terms(proper(X))
     proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
     proper(recip(X)) -> recip(proper(X))
     proper(sqr(X)) -> sqr(proper(X))
     proper(s(X)) -> s(proper(X))
     proper(0()) -> ok(0())
     proper(add(X1,X2)) -> add(proper(X1),proper(X2))
     proper(dbl(X)) -> dbl(proper(X))
     proper(first(X1,X2)) -> first(proper(X1),proper(X2))
     proper(nil()) -> ok(nil())
     proper(half(X)) -> half(proper(X))
     terms(ok(X)) -> ok(terms(X))
     cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
     recip(ok(X)) -> ok(recip(X))
     sqr(ok(X)) -> ok(sqr(X))
     s(ok(X)) -> ok(s(X))
     add(ok(X1),ok(X2)) -> ok(add(X1,X2))
     dbl(ok(X)) -> ok(dbl(X))
     first(ok(X1),ok(X2)) -> ok(first(X1,X2))
     half(ok(X)) -> ok(half(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(add#) = 0
     problem:
      DPs:
       add#(X1,mark(X2)) -> add#(X1,X2)
      TRS:
       active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
       active(sqr(0())) -> mark(0())
       active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
       active(dbl(0())) -> mark(0())
       active(dbl(s(X))) -> mark(s(s(dbl(X))))
       active(add(0(),X)) -> mark(X)
       active(add(s(X),Y)) -> mark(s(add(X,Y)))
       active(first(0(),X)) -> mark(nil())
       active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
       active(half(0())) -> mark(0())
       active(half(s(0()))) -> mark(0())
       active(half(s(s(X)))) -> mark(s(half(X)))
       active(half(dbl(X))) -> mark(X)
       active(terms(X)) -> terms(active(X))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(recip(X)) -> recip(active(X))
       active(sqr(X)) -> sqr(active(X))
       active(s(X)) -> s(active(X))
       active(add(X1,X2)) -> add(active(X1),X2)
       active(add(X1,X2)) -> add(X1,active(X2))
       active(dbl(X)) -> dbl(active(X))
       active(first(X1,X2)) -> first(active(X1),X2)
       active(first(X1,X2)) -> first(X1,active(X2))
       active(half(X)) -> half(active(X))
       terms(mark(X)) -> mark(terms(X))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       recip(mark(X)) -> mark(recip(X))
       sqr(mark(X)) -> mark(sqr(X))
       s(mark(X)) -> mark(s(X))
       add(mark(X1),X2) -> mark(add(X1,X2))
       add(X1,mark(X2)) -> mark(add(X1,X2))
       dbl(mark(X)) -> mark(dbl(X))
       first(mark(X1),X2) -> mark(first(X1,X2))
       first(X1,mark(X2)) -> mark(first(X1,X2))
       half(mark(X)) -> mark(half(X))
       proper(terms(X)) -> terms(proper(X))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(recip(X)) -> recip(proper(X))
       proper(sqr(X)) -> sqr(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(0()) -> ok(0())
       proper(add(X1,X2)) -> add(proper(X1),proper(X2))
       proper(dbl(X)) -> dbl(proper(X))
       proper(first(X1,X2)) -> first(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(half(X)) -> half(proper(X))
       terms(ok(X)) -> ok(terms(X))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       recip(ok(X)) -> ok(recip(X))
       sqr(ok(X)) -> ok(sqr(X))
       s(ok(X)) -> ok(s(X))
       add(ok(X1),ok(X2)) -> ok(add(X1,X2))
       dbl(ok(X)) -> ok(dbl(X))
       first(ok(X1),ok(X2)) -> ok(first(X1,X2))
       half(ok(X)) -> ok(half(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Subterm Criterion Processor:
      simple projection:
       pi(add#) = 1
      problem:
       DPs:
        
       TRS:
        active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
        active(sqr(0())) -> mark(0())
        active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
        active(dbl(0())) -> mark(0())
        active(dbl(s(X))) -> mark(s(s(dbl(X))))
        active(add(0(),X)) -> mark(X)
        active(add(s(X),Y)) -> mark(s(add(X,Y)))
        active(first(0(),X)) -> mark(nil())
        active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
        active(half(0())) -> mark(0())
        active(half(s(0()))) -> mark(0())
        active(half(s(s(X)))) -> mark(s(half(X)))
        active(half(dbl(X))) -> mark(X)
        active(terms(X)) -> terms(active(X))
        active(cons(X1,X2)) -> cons(active(X1),X2)
        active(recip(X)) -> recip(active(X))
        active(sqr(X)) -> sqr(active(X))
        active(s(X)) -> s(active(X))
        active(add(X1,X2)) -> add(active(X1),X2)
        active(add(X1,X2)) -> add(X1,active(X2))
        active(dbl(X)) -> dbl(active(X))
        active(first(X1,X2)) -> first(active(X1),X2)
        active(first(X1,X2)) -> first(X1,active(X2))
        active(half(X)) -> half(active(X))
        terms(mark(X)) -> mark(terms(X))
        cons(mark(X1),X2) -> mark(cons(X1,X2))
        recip(mark(X)) -> mark(recip(X))
        sqr(mark(X)) -> mark(sqr(X))
        s(mark(X)) -> mark(s(X))
        add(mark(X1),X2) -> mark(add(X1,X2))
        add(X1,mark(X2)) -> mark(add(X1,X2))
        dbl(mark(X)) -> mark(dbl(X))
        first(mark(X1),X2) -> mark(first(X1,X2))
        first(X1,mark(X2)) -> mark(first(X1,X2))
        half(mark(X)) -> mark(half(X))
        proper(terms(X)) -> terms(proper(X))
        proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
        proper(recip(X)) -> recip(proper(X))
        proper(sqr(X)) -> sqr(proper(X))
        proper(s(X)) -> s(proper(X))
        proper(0()) -> ok(0())
        proper(add(X1,X2)) -> add(proper(X1),proper(X2))
        proper(dbl(X)) -> dbl(proper(X))
        proper(first(X1,X2)) -> first(proper(X1),proper(X2))
        proper(nil()) -> ok(nil())
        proper(half(X)) -> half(proper(X))
        terms(ok(X)) -> ok(terms(X))
        cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
        recip(ok(X)) -> ok(recip(X))
        sqr(ok(X)) -> ok(sqr(X))
        s(ok(X)) -> ok(s(X))
        add(ok(X1),ok(X2)) -> ok(add(X1,X2))
        dbl(ok(X)) -> ok(dbl(X))
        first(ok(X1),ok(X2)) -> ok(first(X1,X2))
        half(ok(X)) -> ok(half(X))
        top(mark(X)) -> top(proper(X))
        top(ok(X)) -> top(active(X))
      Qed
    
    DPs:
     s#(mark(X)) -> s#(X)
     s#(ok(X)) -> s#(X)
    TRS:
     active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
     active(sqr(0())) -> mark(0())
     active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
     active(dbl(0())) -> mark(0())
     active(dbl(s(X))) -> mark(s(s(dbl(X))))
     active(add(0(),X)) -> mark(X)
     active(add(s(X),Y)) -> mark(s(add(X,Y)))
     active(first(0(),X)) -> mark(nil())
     active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
     active(half(0())) -> mark(0())
     active(half(s(0()))) -> mark(0())
     active(half(s(s(X)))) -> mark(s(half(X)))
     active(half(dbl(X))) -> mark(X)
     active(terms(X)) -> terms(active(X))
     active(cons(X1,X2)) -> cons(active(X1),X2)
     active(recip(X)) -> recip(active(X))
     active(sqr(X)) -> sqr(active(X))
     active(s(X)) -> s(active(X))
     active(add(X1,X2)) -> add(active(X1),X2)
     active(add(X1,X2)) -> add(X1,active(X2))
     active(dbl(X)) -> dbl(active(X))
     active(first(X1,X2)) -> first(active(X1),X2)
     active(first(X1,X2)) -> first(X1,active(X2))
     active(half(X)) -> half(active(X))
     terms(mark(X)) -> mark(terms(X))
     cons(mark(X1),X2) -> mark(cons(X1,X2))
     recip(mark(X)) -> mark(recip(X))
     sqr(mark(X)) -> mark(sqr(X))
     s(mark(X)) -> mark(s(X))
     add(mark(X1),X2) -> mark(add(X1,X2))
     add(X1,mark(X2)) -> mark(add(X1,X2))
     dbl(mark(X)) -> mark(dbl(X))
     first(mark(X1),X2) -> mark(first(X1,X2))
     first(X1,mark(X2)) -> mark(first(X1,X2))
     half(mark(X)) -> mark(half(X))
     proper(terms(X)) -> terms(proper(X))
     proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
     proper(recip(X)) -> recip(proper(X))
     proper(sqr(X)) -> sqr(proper(X))
     proper(s(X)) -> s(proper(X))
     proper(0()) -> ok(0())
     proper(add(X1,X2)) -> add(proper(X1),proper(X2))
     proper(dbl(X)) -> dbl(proper(X))
     proper(first(X1,X2)) -> first(proper(X1),proper(X2))
     proper(nil()) -> ok(nil())
     proper(half(X)) -> half(proper(X))
     terms(ok(X)) -> ok(terms(X))
     cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
     recip(ok(X)) -> ok(recip(X))
     sqr(ok(X)) -> ok(sqr(X))
     s(ok(X)) -> ok(s(X))
     add(ok(X1),ok(X2)) -> ok(add(X1,X2))
     dbl(ok(X)) -> ok(dbl(X))
     first(ok(X1),ok(X2)) -> ok(first(X1,X2))
     half(ok(X)) -> ok(half(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(s#) = 0
     problem:
      DPs:
       
      TRS:
       active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
       active(sqr(0())) -> mark(0())
       active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
       active(dbl(0())) -> mark(0())
       active(dbl(s(X))) -> mark(s(s(dbl(X))))
       active(add(0(),X)) -> mark(X)
       active(add(s(X),Y)) -> mark(s(add(X,Y)))
       active(first(0(),X)) -> mark(nil())
       active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
       active(half(0())) -> mark(0())
       active(half(s(0()))) -> mark(0())
       active(half(s(s(X)))) -> mark(s(half(X)))
       active(half(dbl(X))) -> mark(X)
       active(terms(X)) -> terms(active(X))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(recip(X)) -> recip(active(X))
       active(sqr(X)) -> sqr(active(X))
       active(s(X)) -> s(active(X))
       active(add(X1,X2)) -> add(active(X1),X2)
       active(add(X1,X2)) -> add(X1,active(X2))
       active(dbl(X)) -> dbl(active(X))
       active(first(X1,X2)) -> first(active(X1),X2)
       active(first(X1,X2)) -> first(X1,active(X2))
       active(half(X)) -> half(active(X))
       terms(mark(X)) -> mark(terms(X))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       recip(mark(X)) -> mark(recip(X))
       sqr(mark(X)) -> mark(sqr(X))
       s(mark(X)) -> mark(s(X))
       add(mark(X1),X2) -> mark(add(X1,X2))
       add(X1,mark(X2)) -> mark(add(X1,X2))
       dbl(mark(X)) -> mark(dbl(X))
       first(mark(X1),X2) -> mark(first(X1,X2))
       first(X1,mark(X2)) -> mark(first(X1,X2))
       half(mark(X)) -> mark(half(X))
       proper(terms(X)) -> terms(proper(X))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(recip(X)) -> recip(proper(X))
       proper(sqr(X)) -> sqr(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(0()) -> ok(0())
       proper(add(X1,X2)) -> add(proper(X1),proper(X2))
       proper(dbl(X)) -> dbl(proper(X))
       proper(first(X1,X2)) -> first(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(half(X)) -> half(proper(X))
       terms(ok(X)) -> ok(terms(X))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       recip(ok(X)) -> ok(recip(X))
       sqr(ok(X)) -> ok(sqr(X))
       s(ok(X)) -> ok(s(X))
       add(ok(X1),ok(X2)) -> ok(add(X1,X2))
       dbl(ok(X)) -> ok(dbl(X))
       first(ok(X1),ok(X2)) -> ok(first(X1,X2))
       half(ok(X)) -> ok(half(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     sqr#(mark(X)) -> sqr#(X)
     sqr#(ok(X)) -> sqr#(X)
    TRS:
     active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
     active(sqr(0())) -> mark(0())
     active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
     active(dbl(0())) -> mark(0())
     active(dbl(s(X))) -> mark(s(s(dbl(X))))
     active(add(0(),X)) -> mark(X)
     active(add(s(X),Y)) -> mark(s(add(X,Y)))
     active(first(0(),X)) -> mark(nil())
     active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
     active(half(0())) -> mark(0())
     active(half(s(0()))) -> mark(0())
     active(half(s(s(X)))) -> mark(s(half(X)))
     active(half(dbl(X))) -> mark(X)
     active(terms(X)) -> terms(active(X))
     active(cons(X1,X2)) -> cons(active(X1),X2)
     active(recip(X)) -> recip(active(X))
     active(sqr(X)) -> sqr(active(X))
     active(s(X)) -> s(active(X))
     active(add(X1,X2)) -> add(active(X1),X2)
     active(add(X1,X2)) -> add(X1,active(X2))
     active(dbl(X)) -> dbl(active(X))
     active(first(X1,X2)) -> first(active(X1),X2)
     active(first(X1,X2)) -> first(X1,active(X2))
     active(half(X)) -> half(active(X))
     terms(mark(X)) -> mark(terms(X))
     cons(mark(X1),X2) -> mark(cons(X1,X2))
     recip(mark(X)) -> mark(recip(X))
     sqr(mark(X)) -> mark(sqr(X))
     s(mark(X)) -> mark(s(X))
     add(mark(X1),X2) -> mark(add(X1,X2))
     add(X1,mark(X2)) -> mark(add(X1,X2))
     dbl(mark(X)) -> mark(dbl(X))
     first(mark(X1),X2) -> mark(first(X1,X2))
     first(X1,mark(X2)) -> mark(first(X1,X2))
     half(mark(X)) -> mark(half(X))
     proper(terms(X)) -> terms(proper(X))
     proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
     proper(recip(X)) -> recip(proper(X))
     proper(sqr(X)) -> sqr(proper(X))
     proper(s(X)) -> s(proper(X))
     proper(0()) -> ok(0())
     proper(add(X1,X2)) -> add(proper(X1),proper(X2))
     proper(dbl(X)) -> dbl(proper(X))
     proper(first(X1,X2)) -> first(proper(X1),proper(X2))
     proper(nil()) -> ok(nil())
     proper(half(X)) -> half(proper(X))
     terms(ok(X)) -> ok(terms(X))
     cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
     recip(ok(X)) -> ok(recip(X))
     sqr(ok(X)) -> ok(sqr(X))
     s(ok(X)) -> ok(s(X))
     add(ok(X1),ok(X2)) -> ok(add(X1,X2))
     dbl(ok(X)) -> ok(dbl(X))
     first(ok(X1),ok(X2)) -> ok(first(X1,X2))
     half(ok(X)) -> ok(half(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(sqr#) = 0
     problem:
      DPs:
       
      TRS:
       active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
       active(sqr(0())) -> mark(0())
       active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
       active(dbl(0())) -> mark(0())
       active(dbl(s(X))) -> mark(s(s(dbl(X))))
       active(add(0(),X)) -> mark(X)
       active(add(s(X),Y)) -> mark(s(add(X,Y)))
       active(first(0(),X)) -> mark(nil())
       active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
       active(half(0())) -> mark(0())
       active(half(s(0()))) -> mark(0())
       active(half(s(s(X)))) -> mark(s(half(X)))
       active(half(dbl(X))) -> mark(X)
       active(terms(X)) -> terms(active(X))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(recip(X)) -> recip(active(X))
       active(sqr(X)) -> sqr(active(X))
       active(s(X)) -> s(active(X))
       active(add(X1,X2)) -> add(active(X1),X2)
       active(add(X1,X2)) -> add(X1,active(X2))
       active(dbl(X)) -> dbl(active(X))
       active(first(X1,X2)) -> first(active(X1),X2)
       active(first(X1,X2)) -> first(X1,active(X2))
       active(half(X)) -> half(active(X))
       terms(mark(X)) -> mark(terms(X))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       recip(mark(X)) -> mark(recip(X))
       sqr(mark(X)) -> mark(sqr(X))
       s(mark(X)) -> mark(s(X))
       add(mark(X1),X2) -> mark(add(X1,X2))
       add(X1,mark(X2)) -> mark(add(X1,X2))
       dbl(mark(X)) -> mark(dbl(X))
       first(mark(X1),X2) -> mark(first(X1,X2))
       first(X1,mark(X2)) -> mark(first(X1,X2))
       half(mark(X)) -> mark(half(X))
       proper(terms(X)) -> terms(proper(X))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(recip(X)) -> recip(proper(X))
       proper(sqr(X)) -> sqr(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(0()) -> ok(0())
       proper(add(X1,X2)) -> add(proper(X1),proper(X2))
       proper(dbl(X)) -> dbl(proper(X))
       proper(first(X1,X2)) -> first(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(half(X)) -> half(proper(X))
       terms(ok(X)) -> ok(terms(X))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       recip(ok(X)) -> ok(recip(X))
       sqr(ok(X)) -> ok(sqr(X))
       s(ok(X)) -> ok(s(X))
       add(ok(X1),ok(X2)) -> ok(add(X1,X2))
       dbl(ok(X)) -> ok(dbl(X))
       first(ok(X1),ok(X2)) -> ok(first(X1,X2))
       half(ok(X)) -> ok(half(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     recip#(mark(X)) -> recip#(X)
     recip#(ok(X)) -> recip#(X)
    TRS:
     active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
     active(sqr(0())) -> mark(0())
     active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
     active(dbl(0())) -> mark(0())
     active(dbl(s(X))) -> mark(s(s(dbl(X))))
     active(add(0(),X)) -> mark(X)
     active(add(s(X),Y)) -> mark(s(add(X,Y)))
     active(first(0(),X)) -> mark(nil())
     active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
     active(half(0())) -> mark(0())
     active(half(s(0()))) -> mark(0())
     active(half(s(s(X)))) -> mark(s(half(X)))
     active(half(dbl(X))) -> mark(X)
     active(terms(X)) -> terms(active(X))
     active(cons(X1,X2)) -> cons(active(X1),X2)
     active(recip(X)) -> recip(active(X))
     active(sqr(X)) -> sqr(active(X))
     active(s(X)) -> s(active(X))
     active(add(X1,X2)) -> add(active(X1),X2)
     active(add(X1,X2)) -> add(X1,active(X2))
     active(dbl(X)) -> dbl(active(X))
     active(first(X1,X2)) -> first(active(X1),X2)
     active(first(X1,X2)) -> first(X1,active(X2))
     active(half(X)) -> half(active(X))
     terms(mark(X)) -> mark(terms(X))
     cons(mark(X1),X2) -> mark(cons(X1,X2))
     recip(mark(X)) -> mark(recip(X))
     sqr(mark(X)) -> mark(sqr(X))
     s(mark(X)) -> mark(s(X))
     add(mark(X1),X2) -> mark(add(X1,X2))
     add(X1,mark(X2)) -> mark(add(X1,X2))
     dbl(mark(X)) -> mark(dbl(X))
     first(mark(X1),X2) -> mark(first(X1,X2))
     first(X1,mark(X2)) -> mark(first(X1,X2))
     half(mark(X)) -> mark(half(X))
     proper(terms(X)) -> terms(proper(X))
     proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
     proper(recip(X)) -> recip(proper(X))
     proper(sqr(X)) -> sqr(proper(X))
     proper(s(X)) -> s(proper(X))
     proper(0()) -> ok(0())
     proper(add(X1,X2)) -> add(proper(X1),proper(X2))
     proper(dbl(X)) -> dbl(proper(X))
     proper(first(X1,X2)) -> first(proper(X1),proper(X2))
     proper(nil()) -> ok(nil())
     proper(half(X)) -> half(proper(X))
     terms(ok(X)) -> ok(terms(X))
     cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
     recip(ok(X)) -> ok(recip(X))
     sqr(ok(X)) -> ok(sqr(X))
     s(ok(X)) -> ok(s(X))
     add(ok(X1),ok(X2)) -> ok(add(X1,X2))
     dbl(ok(X)) -> ok(dbl(X))
     first(ok(X1),ok(X2)) -> ok(first(X1,X2))
     half(ok(X)) -> ok(half(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(recip#) = 0
     problem:
      DPs:
       
      TRS:
       active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
       active(sqr(0())) -> mark(0())
       active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
       active(dbl(0())) -> mark(0())
       active(dbl(s(X))) -> mark(s(s(dbl(X))))
       active(add(0(),X)) -> mark(X)
       active(add(s(X),Y)) -> mark(s(add(X,Y)))
       active(first(0(),X)) -> mark(nil())
       active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
       active(half(0())) -> mark(0())
       active(half(s(0()))) -> mark(0())
       active(half(s(s(X)))) -> mark(s(half(X)))
       active(half(dbl(X))) -> mark(X)
       active(terms(X)) -> terms(active(X))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(recip(X)) -> recip(active(X))
       active(sqr(X)) -> sqr(active(X))
       active(s(X)) -> s(active(X))
       active(add(X1,X2)) -> add(active(X1),X2)
       active(add(X1,X2)) -> add(X1,active(X2))
       active(dbl(X)) -> dbl(active(X))
       active(first(X1,X2)) -> first(active(X1),X2)
       active(first(X1,X2)) -> first(X1,active(X2))
       active(half(X)) -> half(active(X))
       terms(mark(X)) -> mark(terms(X))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       recip(mark(X)) -> mark(recip(X))
       sqr(mark(X)) -> mark(sqr(X))
       s(mark(X)) -> mark(s(X))
       add(mark(X1),X2) -> mark(add(X1,X2))
       add(X1,mark(X2)) -> mark(add(X1,X2))
       dbl(mark(X)) -> mark(dbl(X))
       first(mark(X1),X2) -> mark(first(X1,X2))
       first(X1,mark(X2)) -> mark(first(X1,X2))
       half(mark(X)) -> mark(half(X))
       proper(terms(X)) -> terms(proper(X))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(recip(X)) -> recip(proper(X))
       proper(sqr(X)) -> sqr(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(0()) -> ok(0())
       proper(add(X1,X2)) -> add(proper(X1),proper(X2))
       proper(dbl(X)) -> dbl(proper(X))
       proper(first(X1,X2)) -> first(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(half(X)) -> half(proper(X))
       terms(ok(X)) -> ok(terms(X))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       recip(ok(X)) -> ok(recip(X))
       sqr(ok(X)) -> ok(sqr(X))
       s(ok(X)) -> ok(s(X))
       add(ok(X1),ok(X2)) -> ok(add(X1,X2))
       dbl(ok(X)) -> ok(dbl(X))
       first(ok(X1),ok(X2)) -> ok(first(X1,X2))
       half(ok(X)) -> ok(half(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     cons#(mark(X1),X2) -> cons#(X1,X2)
     cons#(ok(X1),ok(X2)) -> cons#(X1,X2)
    TRS:
     active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
     active(sqr(0())) -> mark(0())
     active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
     active(dbl(0())) -> mark(0())
     active(dbl(s(X))) -> mark(s(s(dbl(X))))
     active(add(0(),X)) -> mark(X)
     active(add(s(X),Y)) -> mark(s(add(X,Y)))
     active(first(0(),X)) -> mark(nil())
     active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
     active(half(0())) -> mark(0())
     active(half(s(0()))) -> mark(0())
     active(half(s(s(X)))) -> mark(s(half(X)))
     active(half(dbl(X))) -> mark(X)
     active(terms(X)) -> terms(active(X))
     active(cons(X1,X2)) -> cons(active(X1),X2)
     active(recip(X)) -> recip(active(X))
     active(sqr(X)) -> sqr(active(X))
     active(s(X)) -> s(active(X))
     active(add(X1,X2)) -> add(active(X1),X2)
     active(add(X1,X2)) -> add(X1,active(X2))
     active(dbl(X)) -> dbl(active(X))
     active(first(X1,X2)) -> first(active(X1),X2)
     active(first(X1,X2)) -> first(X1,active(X2))
     active(half(X)) -> half(active(X))
     terms(mark(X)) -> mark(terms(X))
     cons(mark(X1),X2) -> mark(cons(X1,X2))
     recip(mark(X)) -> mark(recip(X))
     sqr(mark(X)) -> mark(sqr(X))
     s(mark(X)) -> mark(s(X))
     add(mark(X1),X2) -> mark(add(X1,X2))
     add(X1,mark(X2)) -> mark(add(X1,X2))
     dbl(mark(X)) -> mark(dbl(X))
     first(mark(X1),X2) -> mark(first(X1,X2))
     first(X1,mark(X2)) -> mark(first(X1,X2))
     half(mark(X)) -> mark(half(X))
     proper(terms(X)) -> terms(proper(X))
     proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
     proper(recip(X)) -> recip(proper(X))
     proper(sqr(X)) -> sqr(proper(X))
     proper(s(X)) -> s(proper(X))
     proper(0()) -> ok(0())
     proper(add(X1,X2)) -> add(proper(X1),proper(X2))
     proper(dbl(X)) -> dbl(proper(X))
     proper(first(X1,X2)) -> first(proper(X1),proper(X2))
     proper(nil()) -> ok(nil())
     proper(half(X)) -> half(proper(X))
     terms(ok(X)) -> ok(terms(X))
     cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
     recip(ok(X)) -> ok(recip(X))
     sqr(ok(X)) -> ok(sqr(X))
     s(ok(X)) -> ok(s(X))
     add(ok(X1),ok(X2)) -> ok(add(X1,X2))
     dbl(ok(X)) -> ok(dbl(X))
     first(ok(X1),ok(X2)) -> ok(first(X1,X2))
     half(ok(X)) -> ok(half(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(cons#) = 0
     problem:
      DPs:
       
      TRS:
       active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
       active(sqr(0())) -> mark(0())
       active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
       active(dbl(0())) -> mark(0())
       active(dbl(s(X))) -> mark(s(s(dbl(X))))
       active(add(0(),X)) -> mark(X)
       active(add(s(X),Y)) -> mark(s(add(X,Y)))
       active(first(0(),X)) -> mark(nil())
       active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
       active(half(0())) -> mark(0())
       active(half(s(0()))) -> mark(0())
       active(half(s(s(X)))) -> mark(s(half(X)))
       active(half(dbl(X))) -> mark(X)
       active(terms(X)) -> terms(active(X))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(recip(X)) -> recip(active(X))
       active(sqr(X)) -> sqr(active(X))
       active(s(X)) -> s(active(X))
       active(add(X1,X2)) -> add(active(X1),X2)
       active(add(X1,X2)) -> add(X1,active(X2))
       active(dbl(X)) -> dbl(active(X))
       active(first(X1,X2)) -> first(active(X1),X2)
       active(first(X1,X2)) -> first(X1,active(X2))
       active(half(X)) -> half(active(X))
       terms(mark(X)) -> mark(terms(X))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       recip(mark(X)) -> mark(recip(X))
       sqr(mark(X)) -> mark(sqr(X))
       s(mark(X)) -> mark(s(X))
       add(mark(X1),X2) -> mark(add(X1,X2))
       add(X1,mark(X2)) -> mark(add(X1,X2))
       dbl(mark(X)) -> mark(dbl(X))
       first(mark(X1),X2) -> mark(first(X1,X2))
       first(X1,mark(X2)) -> mark(first(X1,X2))
       half(mark(X)) -> mark(half(X))
       proper(terms(X)) -> terms(proper(X))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(recip(X)) -> recip(proper(X))
       proper(sqr(X)) -> sqr(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(0()) -> ok(0())
       proper(add(X1,X2)) -> add(proper(X1),proper(X2))
       proper(dbl(X)) -> dbl(proper(X))
       proper(first(X1,X2)) -> first(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(half(X)) -> half(proper(X))
       terms(ok(X)) -> ok(terms(X))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       recip(ok(X)) -> ok(recip(X))
       sqr(ok(X)) -> ok(sqr(X))
       s(ok(X)) -> ok(s(X))
       add(ok(X1),ok(X2)) -> ok(add(X1,X2))
       dbl(ok(X)) -> ok(dbl(X))
       first(ok(X1),ok(X2)) -> ok(first(X1,X2))
       half(ok(X)) -> ok(half(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed
    
    DPs:
     terms#(mark(X)) -> terms#(X)
     terms#(ok(X)) -> terms#(X)
    TRS:
     active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
     active(sqr(0())) -> mark(0())
     active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
     active(dbl(0())) -> mark(0())
     active(dbl(s(X))) -> mark(s(s(dbl(X))))
     active(add(0(),X)) -> mark(X)
     active(add(s(X),Y)) -> mark(s(add(X,Y)))
     active(first(0(),X)) -> mark(nil())
     active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
     active(half(0())) -> mark(0())
     active(half(s(0()))) -> mark(0())
     active(half(s(s(X)))) -> mark(s(half(X)))
     active(half(dbl(X))) -> mark(X)
     active(terms(X)) -> terms(active(X))
     active(cons(X1,X2)) -> cons(active(X1),X2)
     active(recip(X)) -> recip(active(X))
     active(sqr(X)) -> sqr(active(X))
     active(s(X)) -> s(active(X))
     active(add(X1,X2)) -> add(active(X1),X2)
     active(add(X1,X2)) -> add(X1,active(X2))
     active(dbl(X)) -> dbl(active(X))
     active(first(X1,X2)) -> first(active(X1),X2)
     active(first(X1,X2)) -> first(X1,active(X2))
     active(half(X)) -> half(active(X))
     terms(mark(X)) -> mark(terms(X))
     cons(mark(X1),X2) -> mark(cons(X1,X2))
     recip(mark(X)) -> mark(recip(X))
     sqr(mark(X)) -> mark(sqr(X))
     s(mark(X)) -> mark(s(X))
     add(mark(X1),X2) -> mark(add(X1,X2))
     add(X1,mark(X2)) -> mark(add(X1,X2))
     dbl(mark(X)) -> mark(dbl(X))
     first(mark(X1),X2) -> mark(first(X1,X2))
     first(X1,mark(X2)) -> mark(first(X1,X2))
     half(mark(X)) -> mark(half(X))
     proper(terms(X)) -> terms(proper(X))
     proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
     proper(recip(X)) -> recip(proper(X))
     proper(sqr(X)) -> sqr(proper(X))
     proper(s(X)) -> s(proper(X))
     proper(0()) -> ok(0())
     proper(add(X1,X2)) -> add(proper(X1),proper(X2))
     proper(dbl(X)) -> dbl(proper(X))
     proper(first(X1,X2)) -> first(proper(X1),proper(X2))
     proper(nil()) -> ok(nil())
     proper(half(X)) -> half(proper(X))
     terms(ok(X)) -> ok(terms(X))
     cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
     recip(ok(X)) -> ok(recip(X))
     sqr(ok(X)) -> ok(sqr(X))
     s(ok(X)) -> ok(s(X))
     add(ok(X1),ok(X2)) -> ok(add(X1,X2))
     dbl(ok(X)) -> ok(dbl(X))
     first(ok(X1),ok(X2)) -> ok(first(X1,X2))
     half(ok(X)) -> ok(half(X))
     top(mark(X)) -> top(proper(X))
     top(ok(X)) -> top(active(X))
    Subterm Criterion Processor:
     simple projection:
      pi(terms#) = 0
     problem:
      DPs:
       
      TRS:
       active(terms(N)) -> mark(cons(recip(sqr(N)),terms(s(N))))
       active(sqr(0())) -> mark(0())
       active(sqr(s(X))) -> mark(s(add(sqr(X),dbl(X))))
       active(dbl(0())) -> mark(0())
       active(dbl(s(X))) -> mark(s(s(dbl(X))))
       active(add(0(),X)) -> mark(X)
       active(add(s(X),Y)) -> mark(s(add(X,Y)))
       active(first(0(),X)) -> mark(nil())
       active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
       active(half(0())) -> mark(0())
       active(half(s(0()))) -> mark(0())
       active(half(s(s(X)))) -> mark(s(half(X)))
       active(half(dbl(X))) -> mark(X)
       active(terms(X)) -> terms(active(X))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       active(recip(X)) -> recip(active(X))
       active(sqr(X)) -> sqr(active(X))
       active(s(X)) -> s(active(X))
       active(add(X1,X2)) -> add(active(X1),X2)
       active(add(X1,X2)) -> add(X1,active(X2))
       active(dbl(X)) -> dbl(active(X))
       active(first(X1,X2)) -> first(active(X1),X2)
       active(first(X1,X2)) -> first(X1,active(X2))
       active(half(X)) -> half(active(X))
       terms(mark(X)) -> mark(terms(X))
       cons(mark(X1),X2) -> mark(cons(X1,X2))
       recip(mark(X)) -> mark(recip(X))
       sqr(mark(X)) -> mark(sqr(X))
       s(mark(X)) -> mark(s(X))
       add(mark(X1),X2) -> mark(add(X1,X2))
       add(X1,mark(X2)) -> mark(add(X1,X2))
       dbl(mark(X)) -> mark(dbl(X))
       first(mark(X1),X2) -> mark(first(X1,X2))
       first(X1,mark(X2)) -> mark(first(X1,X2))
       half(mark(X)) -> mark(half(X))
       proper(terms(X)) -> terms(proper(X))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
       proper(recip(X)) -> recip(proper(X))
       proper(sqr(X)) -> sqr(proper(X))
       proper(s(X)) -> s(proper(X))
       proper(0()) -> ok(0())
       proper(add(X1,X2)) -> add(proper(X1),proper(X2))
       proper(dbl(X)) -> dbl(proper(X))
       proper(first(X1,X2)) -> first(proper(X1),proper(X2))
       proper(nil()) -> ok(nil())
       proper(half(X)) -> half(proper(X))
       terms(ok(X)) -> ok(terms(X))
       cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
       recip(ok(X)) -> ok(recip(X))
       sqr(ok(X)) -> ok(sqr(X))
       s(ok(X)) -> ok(s(X))
       add(ok(X1),ok(X2)) -> ok(add(X1,X2))
       dbl(ok(X)) -> ok(dbl(X))
       first(ok(X1),ok(X2)) -> ok(first(X1,X2))
       half(ok(X)) -> ok(half(X))
       top(mark(X)) -> top(proper(X))
       top(ok(X)) -> top(active(X))
     Qed